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The dynamic action of gusty winds on long-span bridges

Larose, Guy

Publication date:1997

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Citation (APA):Larose, G. (1997). The dynamic action of gusty winds on long-span bridges. Technical University of Denmark.BYG-Rapport No. R-029

https://orbit.dtu.dk/en/publications/75a957fb-74c3-4ba1-acd9-6c8c0c917a97

BYG∙DT

ISSNISBN 87

Guy L. Larose

The dynamic action of gusty winds on long-span bridges

D A N M A R K S T E K N I S K E UNIVERSITET

RapportU R-029

2002 1601-2917

-7877-088-2

The dynamic action of gusty winds on long-span bridges Guy L. Larose

Department of Civil Engineering DTU-bygning 118 2800 Kgs. Lyngby

http://www.byg.dtu.dk

2002

�

The dynamic action of gusty winds on long�span bridges

Submitted on April �� in partial ful�llment of the requirements for the degreeof Doctor of Philosophy at the Technical University of Denmark�c�Guy L� Larose ��

ii

�

Abstract

The wind loads induced by the bueting action of the turbulence on longspanbridges are the object of this thesis� The study focused on the description of thespatial distribution of the aerodynamic forces for closedbox girder bridge deckswith crosssections that have been designed with emphasis put on aerodynamicsand have� therefore� the appellation �streamlined��

Central to this research is a series of windtunnel experiments that providedan instantaneous image of the wind loading across the span of a twodimensionalmotionless bridge deck in a turbulent ow �eld� The experiments were organised ina parametric study that aimed at de�ning the in uence of the width of the deck inrelation to the scales of the turbulence on the dynamic loading�

The experiments illustrated the importance of the threedimensionality of the liftand overturning forces� The current bueting theory based on the strip assumption did not su�ce to represent the measured wind loading� A threedimensionalanalytical model of the gust loading on a thin airfoil that departs from the stripassumption explained the observed larger spanwise coherence of the forces whencompared to the spanwise coherence of the incident velocity uctuations�

Based on the parametric study� an empirical model of the gust loading on closedbox girder bridge decks was proposed� The empirical formulations included expressions for the aerodynamic admittance as a function of the ratio of the scale of theturbulence to the deck width� The spanwise coherence of the forces was also foundto be a function of the ratio of the turbulence scale to the deck width and a functionof the slenderness of the deck�

iii

�

iv

�

Preface

The research presented here was carried out in the framework of my Ph�D� programme at the Department of Structural Engineering and Materials �BKM� of theTechnical University of Denmark from September �� to April �� The programme was initiated under the hospice of the Danish Research Academy in collaboration with the Danish Maritime Institute �DMI�� The fabrication of the models�the experiments and the data reduction were carried out at DMI while the analysiswas made at BKM�

This thesis is presented as a collection of papers written during the course of myresearch on windstructure interactions� It focuses on the description of the gustloading on motionless bridge decks� Other aspects of wind engineering kept myattention during my Ph�D� studies� namely the eects of wind on bridges duringconstruction and the unique �eld measurements of the windinduced response of a�� m high freestanding pylon for the Storeb�lt East Bridge� The results of theresearch on both topics were reported through two conference papers but are notincluded here�

It was also a pleasure and an inspiration for me to review� in collaboration withNiels Franck� the works of Danish wind engineering pioneers� I learned that the�rst worldwide windtunnel measurements of surface pressure were carried out inCopenhagen almost exactly �� years ago� We wrote a paper on that topic andpresented it at the �th International Conference on Wind Engineering in NewDelhi�India�

My supervision committee was composed of Prof� Niels J� Gimsing� Dr� Cla�es Dyrbye and Prof� Hiroshi Tanaka� Their knowledgeable guidance is most sincerely acknowledged as well as their clairvoyance in encouraging me to learn about wind ona sailboat across the Atlantic Ocean�

I would like also to express my grateful thanks to the COWIfoundation for the�nancial support that made possible the experimental part of this research� Acknowledgements are extended to DMI�s management� in particular Aage Damsgaard� who encouraged and made room for this study� and to my colleagues at DMIand BKM who kindly coped with me� The invaluable discussions and contributionsof my friends and coauthors Jakob Mann from Ris� and Flora M� Livesey fromDMI are warmly acknowledged�

This is the second edition of my Ph�D� thesis� It includes the revisions suggestedby the members of my examining committee composed of Professors Erik Hjorth

v

Hansen and Steen Krenk� and Dr Allan Larsen� Their comments and suggestionsare greatly acknowledged� The �rst edition was submitted on April �� inpartial full�llment of the requirements for the degree of Doctor of Philosophy at theTechnical University of Denmark�

En�n� mille mercis �a Simone et Emil pour leur patience l�egendaire et leurs multiples encouragements ainsi qu��a Flora sans qui la vie ne serait pas la m�eme�

Cette th�ese est d�edi�ee �a ma famille qui m�est tr�es ch�ere et qui ne cesse de grandir�

GLL� Lyngby� Oct� ��

vi

Contents

Abstract iii

Preface v

Table of contents vii

Nomenclature xi

General introduction �

Part I BACKGROUND

Part I�A Prediction of the bu�eting response

� Introduction �

� Theoretical background �

�� De�nition � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Linearised equation of motion � � � � � � � � � � � � � � � � � � � � � � �

� The spectral approach ��

�� Assumptions and limitations � � � � � � � � � � � � � � � � � � � � � � ��

�� Bueting analysis� lift � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Simpli�ed expression� resonant modal response � � � � � � � � � � � � ��

�� Shortcomings � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Improvements � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Predictions in the time domain� a review ��

�� Quasisteady aerodynamics � � � � � � � � � � � � � � � � � � � � � � � ��

�� Corrected quasisteady aerodynamics � � � � � � � � � � � � � � � � � � ��

�� Indicial functions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Rational function approximations � � � � � � � � � � � � � � � � � � � � ��

�� Equivalent oscillator � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Experimental approaches ��

�� Section model method � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Predictions of fullscale response � � � � � � � � � � � � � � � � � � � � ��

�� Threedimensional physical models � � � � � � � � � � � � � � � � � � � ��

References ��

vii

Part I�B The span�wise coherence of wind forces

� Joint acceptance function and spanwise coherence ��

� Flow over a bridge deck with sharp edges �

� Some e�ects of the deck on the turbulence ��

� Empirical models of spanwise coherence ��

�� Models based on the strip assumption � � � � � � � � � � � � � � � � � ��

�� When the strip assumption is not valid � � � � � � � � � � � � � � � � � ��

� Concluding remarks �

References �

Part II EXPERIMENTS AND ANALISYS

Part II�A Direct measurements of bu�eting windforces on bridge decks

� Introduction ��

� Description of the experiments ��

�� Instantaneous force measurements � � � � � � � � � � � � � � � � � � � ��

�� Flow investigations � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Analysis of the ow conditions ��

�� Isotropic turbulence model and the von K�arm�an spectrum � � � � � � ��

�� Twopoint statistics� the spanwise coherence � � � � � � � � � � � � � ��

�� Least square �t of wind data � � � � � � � � � � � � � � � � � � � � � � ��

�� The FichtlMcVehil spectrum � � � � � � � � � � � � � � � � � � � � � � ��

� Characteristics of the unsteady wind forces �

�� Crosssectional forces and aerodynamic admittance � � � � � � � � � � ��

�� Spanwise crosscorrelation and cocoherence � � � � � � � � � � � � � ��

�� Crosscorrelation coe�cient � � � � � � � � � � � � � � � � � � � ��

�� Cocoherence� wind velocity versus forces � � � � � � � � � � � ��

�� In uence of the Lw�B ratio � � � � � � � � � � � � � � � � � � � ��

�� In uence of the railings � � � � � � � � � � � � � � � � � � � � � ��

�� In uence of the angle of wind incidence � � � � � � � � � � � � ��

viii

� Concluding remarks ��

References ��

Appendix A Photographs of the experimental setup ��

Part II�B Gust loading on streamlined bridge decks


� Theory� lift force induced by turbulence ��

�� Quasisteady aerodynamics � � � � � � � � � � � � � � � � � � � � � � � ��

�� Introducing unsteadiness � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Lifting the strip assumption � � � � � � � � � � � � � � � � � � � � � � ��

� Experimental validations ��

�� The experiments � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Description of the incident turbulent ow �eld � � � � � � � � ��

�� Measured aerodynamic admittance functions � � � � � � � � � ��

�� Measured spanwise cocoherence of the forces � � � � � � � � ��

�� Applications of the �D analytical model � � � � � � � � � � � � � � � � ��

�� Discussions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Empirical model of the spatial distribution of the bu�eting forceson closedbox girder bridge decks ��

�� Crosssectional aerodynamic admittance as a function of Lw�B � � � ��

�� Model of the spanwise cocoherence of the aerodynamic forces � � � ��

� Conclusions ��

References ��

Part III APPLICATIONS

Part III�A Experimental determination of the aero�dynamic admittance of a bridge deck segment


� Description of the approach ��

ix

� Experimental veri�cation ��

�� Force measurements � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Evaluation of the joint acceptance function � � � � � � � � � � � � � � �� The aerodynamic admittances � � � � � � � � � � � � � � � � � � � � � � ��

� Truss girder decks and other blu� crosssections ��

� Conclusion ��

References ��

Part III�B Performance of streamlined bridge decksin relation to the aerodynamics of a �at plate


� Comparison of static force coe�cients ��

� Aerodynamic derivatives and admittances ��

� Prediction of the bu�eting response ��

� Conclusions ��

References ��

x

Nomenclature

jA�f��j aerodynamic admittance function or AAFA�� dimensionless aerodynamic derivatives associated with the tor

sional degree of freedom of a bridge deckB deck widthBx�z�� subscript� background or quasisteady component of the dynamic

response in the given degree of freedomc� c�� c�� c� exponential decay constants for spanwise cocoherence of the

wind uctuations and aerodynamic forcescocohw�L spanwise cocoherence �or normalized cospectrum� of the w com

ponent of the wind or the vertical aerodynamic forces L

coh��w�L spanwise root coherence of the w component of the wind velocity

or the vertical aerodynamic forcesCm steady moment coe�cient in the torsional direction� torsion about

the bridge axis� based on B�

Cx steady bodyforce coe�cient in the horizontal direction� based onB

Cz steady bodyforce coe�cient in the vertical direction� based on BC �m�x�z � dCm�x�z�d�� rate of change �slope� of the respective steady

force coe�cients with angle of wind incidence � in radiansCp pressure coe�cientCo�� cospectrum� i�e� real �inphase� component of the cross spectrum

between points � and �D depth of bridge deckFm�x�z aerodynamic force per unit length in the given degrees of freedomf frequencyfx�z�� frequency of �rst symmetric mode of vibration respectively in the

lateral� vertical and torsional degrees of freedom� in still airfj natural frequency of mode j in still airf� � fB�V � reduced frequencyf�j reduced frequency based on the natural frequency of mode j

fm frequency associated with the FichtlMcVehil spectrumGx�z�� damping coe�cient in the lateral� vertical and torsional directionG�j generalized damping coe�cient for mode j

Gcrit critical dampingg gravitational acceleration or statistical peak factorH�

�� dimensionless aerodynamic derivatives associated with the vertical degree of freedom of a bridge deck

Iu�v�w longitudinal� transversal and vertical turbulence intensity�respectively

xi

I� mass moment of inertia per unit length of a bridge deckJ�f�� joint acceptance function or JAFJ�� J� Bessel functionsK�

j generalized stiness of the deck in mode j

K�� modi�ed Bessel functions of the second kind

k� � ��f� �V � wave number k� k�B��L length scale associated with the von K�arm�an spectrumLx�y�zu�v�w horizontal� transversal and vertical integral length scales of tur

bulence in the x� y� z directionl span lengthL length scale of turbulence corresponding to the inverse of the wave

number at which the wind spectrum has its maximumM�

j generalized mass of mode j

m mass of the deck per unit lengthp�� probability distribution of angle �Qu�� quadrature spectrum i�e� imaginary �outofphase� component of

the cross spectrum between points � and �R�� crosscorrelation coe�cient between points � and �Rx�z�� subscript� resonant component of the dynamic response in the

given degree of freedomr peakness parameter of the FichtlMcVehil spectrum�r mean response r rootmeansquare response�r expected peak response rRj

resonant rootmeansquare modal responseSu�v�w spectra of the uctuations of the wind velocity components� lon

gitudinal� transversal and verticalSFz spectrum of the vertical wind load on a bridge spanSL spectrum of the lift force on a chordwise strip of the deckSL�L� crossspectrum of the lift forces between point � and �t timeu longitudinal component of the wind velocityV wind velocity in the alongwind direction�V mean wind velocityVr apparent velocity of the windV � � V�fB� reduced velocityv transversal component of wind velocityw vertical component of wind velocity or angular velocity

xii

x horizontal direction� perpendicular to the bridge axis!x �rst derivative of horizontal motion with respect to time �velocity�y spanwise or transversal directionz vertical direction!z �rst derivative of vertical motion with respect to time �velocity�� instantaneous angle of wind incidence"�x mean horizontal amplitude of the deck motion"y spanwise separation between points y� and y�� modi�ed von K�arm�an parameter�a�� normalized aerodynamic stiness in torsion# Gamma function� von K�arm�an parameter�j mode shape of a given mode j cycling ratej angular frequency of vibration of a given mode j��f�� theoretical aerodynamic admittance function� air density B� Rj

rootmeansquare background and resonant response u�v�w rootmeansquare of wind velocity uctuations� designates torsional degree of freedom or rotation of the deck

about the bridge axis�aj aerodynamic damping as a fraction of critical for mode j�sj structural damping as a fraction of critical for mode j

xiii

�

xiv

The dynamic action of gusty winds �

General introduction

Objective and impetus for research

The knowledge of the spatial distribution of the forces induced by the gustiness ofthe wind is at the heart of the predictions of the bueting response of longspanbridges� The objective of this work is to describe the spatial distribution of theseforces on a motionless closedbox girder bridge deck�

The impetus for the research came from an analysis of the error margin of thecurrent prediction methods of bridge bueting� In the early phase of this study�a review of the comparisons found in the literature between analytical predictionsof the bueting response and measured response in full scale or model scale wascarried out� It revealed that in some instances the error margin of the analyticalpredictions was found to be small while in other cases� the analytical predictionsgreatly overestimated or underestimated the measured response�

This apparent inconsistency suggested that a more indepth study of each casebe conducted to arrive at a reliable assessment of the error margin since manyparameters could in uence the comparisons� For example� the description of theincident wind �eld in fullscale is an important source of uncertainty� let alone itsmodelling in a wind tunnel� Also� the evaluation of the structural damping and itsvariability with amplitude of motion is another major parameter that in uences thecomparisons�

Often� however� the background information is not su�cient to conduct such indepth studies� leaving room for interpretation and doubts� This brought the authorto perform a review of the shortcomings of the analytical methods of predicitons ofthe bueting of longspan bridges which is presented in Part I of this thesis� Thereview indicated that the modelling of the action of the turbulence on closedboxgirder bridge decks was not clear� This is especially true when the bridge deck hasa characteristic length �e�g� width� similar to the length scales of the incident gusts�This is the case for the majority of modern closedbox girder bridge decks with deckwidth ranging from �� to �� m immersed in atmospheric boundary layer winds with�� m long gusts but with vertical and lateral length scales not larger than ��to �� m�

The analytical calculations of bridge bueting� either in the frequency domain orin the time domain� are generaly based on the strip assumption despite the fact thatthis assumption is valid only when the incident gusts have much larger scales thanthe characteristic length of the bridge deck� Based on the strip assumption� thespanwise coherence of the aerodynamic forces can be represented by the spanwisecoherence of the incident wind velocity uctuations� The experiments carried out inthe present research programme and reported in full in Part IIA of this thesis aimedat de�ning the error margin of the calculations of the vertical and torsional gust

� G� L� Larose

loading on a motionless bridge deck based on the strip assumption� The in uenceof the ratio between the size of the gusts and the deck width was also studied bysystematically changing both quantities in turn�

The distinction �motionless� is made in an attempt to dissociate the aerodynamicforces due uniquely to the incident wind turbulence from the motioninduced forcesoften associated with the wake of the bridge deck� It has been observed that thespatial distribution of the wind loading can dier importantly if the twodimensionalbody is in motion or not and if this motion is organised or not� Here the emphasisis made on the forces induced by the wind gustiness only and any distortion andreorganisation of the turbulence that could occur if the bridge deck would undergorelatively important oscillations in term of amplitudes is not considered�

This research is a logical progression of the work carried out by the author in theframework of a M� Eng� Sc� thesis at The Boundary Layer Wind Tunnel Laboratoryof The University of Western Ontario� Canada� During that earlier work� the evaluation of the spanwise distribution of the lift and torsional forces on a closedboxgirder bridge deck based on the strip assumption led to an underestimation of theforces when compared to direct measurements in a wind tunnel� The lift crosssectional admittance was found� however� to be lower than predicted by the lineartheory for a thin airfoil in a fully correlated gust� The crosssection studied was thedeck of the Storeb�lt East Bridge�

In the present research� a parametric study of the in uence of the width of thedeck and of the turbulence scales on the dynamic wind loading was made by keepingconstant the deck depth and the edge con�guration but varying the deck width of theexperimental model� The edge con�guration selected is similar to the con�gurationof the Storeb�lt East Bridge�

A secondary objective of this research was to quantify the relationships betweenthe scales of turbulence� the slenderness of the deck� and the gust loading� Thisanalysis was performed to de�ne a model of the loading that departs from the stripassumption and reduces the error margin of the bueting predictions�

Synopsis

This thesis is divided into three parts that followed closely the pattern adoptedduring this research� Part �� Background exposes the theoretical backgroundfor the research and presents the impetus for the research� Part �� Experimentsand analysis presents a detailed account of the experimental work carried out inthe framework of this research� It also presents the analytical treatment of thedata followed by the de�nition of a model of the bueting wind loading� Part ��Applications presents some applications of the proposed model� Each part wassubdivided into two smaller parts which are presented in the form of standalonepapers with their own introduction� development and conclusions�

The dynamic action of gusty winds �

Initially a review of the current methods of prediction of the bueting responseof bridge decks was made� with emphasis on the limitations and shortcomings ofthe spectral approach� A summary of this review is given in Part IA� with somesuggestions for improvements of the analytical predictions� In Part IB� a detailedstateoftheart description of the spatial distribution of the aerodynamic forces onclosedbox girder bridge decks is given� This part was presented at an internationalcolloquium on blubody aerodynamics during the course of the study and waspublished in the proceedings�

Part IB sets the stage for the planning and execution of the experiments described in Part IIA� The main �ndings of the parametric study are also given inPart IIA� Part IIB focused on the analysis of the results and the buildingup ofempirical models of the crosssectional admittances and spanwise coherence of theaerodynamic forces for the family of closedbox girder bridge decks of this study�The empirical formulations are based on a threedimensional ��D� analytical modelof the gust loading on a thin airfoil that departs from the strip assumption�

This �D analytical model of the gust loading is presented in Part IIB alongwith examples of calculations for isotropic turbulence� Part IIB brings forward aphysical and analytical explanation for previously reported observations regardingthe variations of the crosssectional admittance of the lift forces and its spanwisedistribution as a function of reduced frequency and length scales of turbulence� Thecontents of both Part IIA and Part IIB were published in summarised form in tworefereed journal papers�

In Part III� some applications of the models of Part IIB are reported� A methodof evaluation of the admittance of bridge decks of any crosssection is described inPart IIIA� The quantity measured has a �D character and was termed aerodynamicadmittance of a segment �segmental admittance� as opposed to crosssectional admittance since it includes in its de�nition a portion of the spanwise coherence ofthe aerodynamic forces� The validation of the method was made possible throughthe empirical model presented in Part IIB�

Predictions of the bueting response for various closedbox girder bridge decksare given in Part IIIB and are compared to bueting predictions for a at plate�The analytical calculations are based on the theory presented in Part IA with theimprovements suggested by Part IIB and the experimental technique presented inPart IIIA for the determination of the aerodynamic admittance� Part IIIA andPart IIIB were published in refereed journals�


�

Part I�A Prediction of the bu�eting response �

Part I

Background

Part I�A

Prediction of the bu�eting response

Guy L� Larosea�b

a Department of Structural Engineering and Materials� Technical University of Den�

mark� �� Lyngby� Denmarkb Danish Maritime Institute� Hjortek�rsvej � �� Lyngby� Denmark

Abstract

A review of the methods of prediction of the bueting response of longspan bridgesand other linelike structures is presented� Analytical and experimental approachesare discussed� with emphasis on the spectral approach� Limitations� shortcomingsand suggestions for improvements of the bueting theory are summarized�

� Introduction

Longspan bridges must be designed to withstand the static drag forces inducedby the timeaveraged wind pressure� If not properly designed� the drag load andthe selfexcited aerodynamic moment on the deck section can cause static buckling�lateral buckling and torsional divergence�� In addition� bridges may respond dynamically to the eects of the wind ow over the deck section� This windinduceddynamic response is generally classi�ed into three major categories� depending onthe mechanisms involved�

� random response due to bueting by turbulence$

� vortexinduced response$ and

� aerodynamic instability�


For the prototype structure� not only more than one of these types of responseoccur at the same time� but also the dierent modes of oscillation of the structuremay be susceptible to dierent excitation� Wind tunnel tests carried out in smoothand turbulent ow on twodimensional section models in dynamic systems withadjustable frequency and damping or on threedimensional aeroelastic models allowthe study of these three categories of response�

Here� only the response due to bueting by turbulent wind will be described�with emphasis on the prediction of the full scale behaviour based on the results ofthe section model tests combined with the spectral approach�

� Theoretical background

�� De�nition

Bueting is the random response of the bridge due to wind forces associated withthe pressure uctuations on the bridge deck caused by gustiness of the wind owover the section� Since this gustiness has vertical as well as horizontal variationsin velocity� the analysis must include the eects of a random variation in the angleof attack� Bueting normally increases monotonically with the mean velocity andis thus more important at maximum winds� It is also a function of the turbulenceintensity of the oncoming ow and of the structural damping of the bridge�

�� Linearised equation of motion

The prediction of the bueting response to turbulent wind is generally secondaryto the question of aerodynamic stability� However� when the bridge is shown tobe stable� the question of the response to gusts is important for the design of thesuperstructure and the assessment of the user comfort by predicting the accelerationlevel� The theoretical approach developed by Davenport in �� %�� & is the resultof the �rst thorough analytical investigation of the problem and is the most widelyused method of prediction of the eects of gusts on civil engineering structures�Sections �� and �� give a summary of this approach�

The derivation of the theoretical expressions for the prediction of the responseof a bridge deck to turbulent wind� assuming quasisteady aerodynamics� has beendescribed in detail elsewhere %�&� Only the main expressions are reproduced here�focusing on the vertical degree of freedom�

Assuming quasisteady aerodynamics� and referring to Fig� �� the vector ofthe velocity of the wind relative to the deck in motion �or the apparent velocity�can be expressed by�

Vr �q�w � !z�� ' ��V ' u� !x��


Figure �� Notation for deck motion

V �r � w� � �w !z ' !z� ' �V � ' �V u ��

� �V !x' u �V ' u� � u !x� !x �V � !xu' !x��

where u and w are respectively the longitudinal and the vertical components of thewind uctuations� �V is the mean wind speed� x and z and their time derivative!x and !z denote in turns displacement and velocity of the deck in the lateral andvertical degrees of freedom�

For small displacements and small velocity uctuations� the terms involving theproducts of w� u� !x� !z can be neglected and the above expression can be reduced to�

V �r � �V � ' ��V u� � �V !x� ��

The apparent instantaneous angle of attack is given by�

� � � 'w � !z � nB !��V ' u� !x

��

where n is a factor representing the position of the point of application of thevertical aerodynamic forces that produce a torsional motion� For a thin airfoil atlow reduced frequency� n � �� i�e� the equivalent point of application of the liftforces is a quarter of the chord� B� upstream of the torsional centre� For bridge decksthe quasisteady values of n is often taken as � �� but in practice it varies withreduced frequency� The term nB !� was in fact originally introduced for bridge decksby Irwin %�& where n was kept as a oating paramter� to express quasistatically thecontribution of the torsional aerodynamic damping in the equation of motion�

De�ning the lift coe�cient Cz as a linear function of the angle of wind incidence� we have�

Cz�� Cz� ' C �

z� ��


where C �z is the rate of change of the lift coe�cient with angle of wind incidence in

radians� Note that Cz is the lift or vertical force coe�cient for a coordinate system�xed to the bridge deck� and Cz� is the linearised lift coe�cient at � � �� The liftforce per unit length is de�ned by�

Fz � Cz��

��V �

r �B ��

Fz ��Cz� ' C �

z��

��B� �V � ' ��V u� � �V !x�

��

�

�Cz� ' C �

z

�� '

w � !z � nB !��V ' u� !x

��

��B �V � ' �B �V u� �B �V !x

��

Neglecting the terms with the product of u� w and !z� !x� equation �� becomes�

Fz ��B

�

�Cz�

�V � ' �Cz��V u' C �

zw�V � C �

z !z�V � �Cz�

�V !x' C �

z��V � � C �

znB!� �V

��

��The �rst term represents the mean wind load� the second and third terms rep

resent the bueting wind load due to� respectively� the alongwind gusts and thevertical gusts� the fourth term is a motioninduced wind load and represents anaerodynamic damping which reduces the dynamic loading if C �

z is positive� and thelast terms represent the wind load due to coupling between alongwind� verticaland torsional motions� Similar formulations can be derived for the drag force andpitching moment�

The wind loads due to the bueting action of the wind �subscript b� are expressedby�

Fx�b �� V B

�

�Cxu' C �

xw

��

Fz�b �� V B

�

�Czu' C �

zw

��

M��b �� V B�

�

�Cmu' C �

mw� ��

Combining the load uctuations from the wind with the equations of motion ofthe deck� the quasisteady equations for motion of a suspended deck in gusty windcan be expressed as follows for the three main degrees of freedom %�&�

Mx�x' �Gx ' � �V BCx� !x'Kxx'

��

�� V BC �

x !z ��

�� V �BC �

x�

�� Fx�b ��

Mz�z'�Gz'�

�� V BC �

z� !z'Kzz'

�� V BCz !x� �

�� V �BC �

z� ��

�� V nB !�

�� Fz�b ��


I��'�G�'�

��B�C �

mn �V � !�'�K��

�� V �B�C �

m��'

�� V B�Cm !x'

�

�� V B�C �

m !z

�� M��b

��where Mx�z and I� represent the eective inertias of the deck for the horizontal�vertical and torsional motions� Gx�z�� represent the eective viscous damping andKx�z�� the eective stiness� The terms in % & are coupling terms� and all productsof a force coe�cient and a position or velocity represent motioninduced forces �orselfexcited forces��

The importance of the steady aerodynamic force coe�cients can be depicted fromthis set of equations�

� Cx and C �z contribute to the damping in drag and lift respectively by an

amount proportional to the mean wind velocity$

� since Cx is always positive for small angles of wind incidence� its contributionto the horizontal damping will always be positive$

� the vertical aerodynamic damping is proportional to C �z� If this slope is neg

ative� the overall vertical damping will be negative for a certain wind speedand galloping instability will develop$

� the pitching moment coe�cient C �m contributes to the torsional stiness of

the system� If it is positive� the torsional stiness will decrease in proportionto V �� The torsional frequency will thus decrease and at some wind velocityit will approach the vertical frequency creating grounds for development ofclassical utter� At some other velocity the eective torsional stiness willbecome negative and this time divergent instability will arise�

A steep pitching moment slope C �m would mean a fast reduction of the eective

torsional stiness and would likely induce larger torsional forces due to verticalgusts$

� aerodynamic coupling terms are present for the three degrees of freedom�The terms in Cm and C �

x will be negligible for a streamlined bridge deck sincethese aerodynamic terms evaluated at zero incidence are generally small� Thismeans practically no coupling in the horizontal motion� Coupling will occuronly if two or three modes of vibration in dierent degrees of freedom have asimilar frequency and mode shape$

� the gust forces depend on the longitudinal and vertical component of the uctuating velocity� u and w� The contribution of the longitudinal componentis a function of the aerodynamic coe�cients at zero incidence which are inmost cases relatively small except for the horizontal direction where the slopeC �x is even smaller$

�� G� L� Larose

� a steep slope for C �z would provide strong vertical aerodynamic damping but

would also induce larger vertical forces due to the vertical gusts w�

This set of equations does not include unsteady aerodynamic coe�cients thatrepresent the outofphase components of the aerodynamic forces and contribute tothe aerodynamic damping and stiness of the system� The unsteady coe�cientscan be introduced when measurements of motional aerodynamic derivatives havebeen performed�

The equations above can represent aerodynamically coupled motions� However�the coupling terms are often small and therefore neglected� Also the contributionof the aerodynamic stiness is often considered negligible in comparison to thestiness of the bridge itself� This leaves the aerodynamic damping as the mostsigni�cant aerodynamic force induced by motion� If it is negative and greater thanthe structural damping� large motion will result� The response of each uncoupledmode of vibration to the turbulent wind can be studied separately and superimposedat the end$ usually� only the lowest modes are signi�cant�

� The spectral approach

A review of what is known as Davenport�s spectral approach for the prediction ofthe bueting response of a bridge deck is presented in this section� It is a statisticalapproach� based on work by Liepmann %�&� which de�nes the wind loading as astochastic process of the stationary random type�

Power spectra can thus be used to describe the stochastic loading as well as thestatistical properties of the turbulence� The mathematical formulation is thereforesimple since it can be presented as a product of linear functions of frequency orreduced frequency f��

It relies heavily on the aerodynamic properties of the deck crosssection obtainedfrom wind tunnel tests on section models in turbulent ow� The more aerodynamicinformation about the crosssection studied� the better the predictions� However�the advantages of the frequency domain formulation are numerous� since� for example� all the unsteady characteristics of the deck are better described in the frequencydomain and are extracted frequency by frequency from experiments� It is also wellsuited for predictions of extreme responses�

�� Assumptions and limitations

The assumptions and limitations of the original formulation are summarized asfollows�

� wind loading due to gusts is a stochastic process of the stationary randomtype$

Part I�A Prediction of the bu�eting response ��

� quasisteady assumption� the instantaneous forces on the deck are taken tobe equal to the steady forces induced by a steady wind having the samerelative velocity and direction as the instantaneous wind� The steady forcecoe�cients of the deck crosssection and their variations with angle of attackare thus considered su�cient to solve the equations of motion of the deck inturbulent wind as described in the previous section$

� strip assumption� the aerodynamic forces on one strip are only due to theincident wind uctuations on that strip� The spatial description of the wind uctuations can then be taken as representative of the spatial distribution ofthe bueting loading$

� the natural frequencies of the mode of vibrations of the structure are wellseparated so that coupling can be neglected� The analysis can thus be donemode by mode and the response can be determined by a superposition of aseries of single degreeoffreedom systems$

� The turbulent uctuations are assumed small enough compared to the meanvelocity so that gust loads can be expressed as linear functions of the gustvelocities� Also� the variation of the static coe�cients around �� is taken asbeing linear$

� the worst wind direction is assumed to be wind normal to the bridge axis andthe mean angle of wind incidence is taken as ��$

� crossspectra between the u and w components of the wind are negligible$

� aerodynamic damping varies linearly with wind speed and is independent ofamplitude� The contribution of the aerodynamic stiness is neglected$

� and� the parent distribution of the extreme response is assumed to be Gaussian�

This approach is most valid for small reduced frequency f� � fB�V �or largereduced velocity� where the time taken for the ow to traverse the bridge deck isvery short compared to the oscillation time or� in other words� the eects of themotion of the deck are communicated rapidly to the ow region surrounding it�In another sense� it is valid when disturbances in the ow have appreciably largerdimensions than the deck itself�

The limitations of the quasisteady aerodynamic assumption can be lifted using the notion of aerodynamic admittance as �rst proposed by Sears and used byLiepmann %�& �a thorough discussion of this aspect as well as the use of the strip assumption is presented in Part IIB�� Davenport suggested applying the aerodynamicadmittance %�& to represent� i� the loss of lift for the higher frequency componentsof the turbulence �the small gusts�$ and� ii� the spatial variations in the ow since


the forces are not necessarily due to the wind uctuations at one point but on aregion surrounding a chordwise strip�

Davenport also suggested in %�& the use of unsteady aerodynamic coe�cientsdetermined from the inphase and outofphase components of the aerodynamicforces to evaluate the aerodynamic damping forces� This was introduced in thebueting formulation at a later stage by Irwin %�& and Scanlan %�&�

�� Bu�eting analysis� lift

Assuming that the bueting loading is a stationary random process� equation ��can be transformed to the frequency domain�

SL�f��

�� V B

�

�� C�

z Su�f�� jAu�z�f

��j� ' C �

z�Sw�f

�� jAw�z�f��j�

��

where SL�f�� is the spectrum of the lift force per unit length on a crosssectional

strip of the deck �also referred to as pointlike load�� f� is the reduced frequency� fB� �V � jAu�w�z�f

��j� are the lift aerodynamic admittances due to the u or wcomponents of the wind turbulence� and Su�w�f

�� are the spectral densities of theu and w components of the wind�

The aerodynamic admittances can either be measured� evaluated or approximatedusing analytical expressions derived for a thin airfoil �as described in Part IIB��Liepmann�s approximation to the Sears� function is the most commonly used formfor the lift aerodynamic admittance for a thin airfoil %�&�

j �z�f�� j��

� ' ��f��

This approximation is within ��( of the more evolved Sears� function and is preferred to the latter due to its simplicity and since it has shown in some occasionsto represent qualitatively direct measurements on bridge decks�

For drag� Irwin has derived an analytical expression for the admittance of theu component which varies as a function of the ratio between the scales of theturbulence and the size of the bridge deck %�&�

In practice it is di�cult to distinguish between the eects of u and w so theadmittances are generally lumped and �� becomes�

SL�f��

�� V B

�

��

jAz�f��j�

��C�

zSu�f�� ' C �

z�Sw�f

��

From pointlike to linelike� The transition from pointlike load to load on aspan l or linelike load is made via the joint acceptance function Jz�f

�j ��

SFz�f�

j � � SL�f�� j Jz�f�j � j� ��


where

jJz�f�j �j� �

Z l

�

Z l

�

SL�L��"y� f��

SL�f��j�y�� j�y�� dy�dy��

�j is the jth mode shape and SL�L� is the crossspectrum of the lift force betweenstrips � and � separated by "y�

The natural wind is composed of gusts that can be visualized as cigars� e�g� ��m long and �� m in diameter� being transported and streched by the mean ow�eld� These gusts will pass by the structure and generate bueting forces� Given a�� m bridge span for example� it is unlikely that a series of large gusts will pass bythe entire span at the same time� The bueting forces are thus not fully correlatedspanwise� Also� the gust loading pattern will aect the structure in a dierentway for each dierent mode of vibration� The joint acceptance function takes theseeects into account� It measures the correlation between the spatial distribution ofthe forces across the span and the mode� There is one joint acceptance function permode of vibration of the deck�

The evaluation of the crossspectrum of the lift forces is di�cult� but under thebasis of the strip assumption�

SL�L��"y� f��

SL�f�� Sw�w��"y� f��

Sw�f�� coh��w �"y� f��

It has been reported that �� might not be valid for closedbox girder bridgedecks since their width is often of the same dimension as the scales of the w component of the turbulence� The evaluation of the spanwise root coherence of the

aerodynamic forces coh��L �"y� f�� is thus at the heart of the de�nition of the wind

loading on a span� Part IIA and Part IIB of this thesis deals with its experimentaland analytical evaluation�

Spectrum of the bu�eting response� The spectrum of the response of a givenmode to the bueting forcing described above can be calculated using$

Srz�f�

j � � SFz�f�

j � jH�f�j �j� ��

where H�f�j � is the single degreeoffreedom mechanical admittance function ofmode j� H�f�j � is a function of reduced frequency and damping� both being in uenced by the aerodynamic forcing� This in uence is represented by adding thecontribution of the aerodynamic damping �a to the structural damping �s and alsoby correcting the frequency term to take into account the aerodynamic stiness�The latter correction often has negligible in uence on the bueting response and isthus omitted in the following� H�f�j � can be expressed by�

j H�f�j � j��

�� f�

f�j��

'

��s�j ' �a�j�

f�

f�j

��


The mean square vertical response of mode j is�

�rj �

Z�

�SFz�f

�

j � jH�f�j �j�df��

and the mean square response for all modes at spanwise position y is�

�r�y� �nX

j�

�rj ��j �y��

Background and resonant components� The dynamic response can be divided in its background and resonant components�

�Background �

Z�

�f�SL�f

�� j Jz�f�� j� d ln f� ��

�Resonantj � f�j SL�f�

j � j Jz�f�j � j��

�sj ' �a�f�j �

� ��

where the last two terms of equation �� are an approximation �error of less than'��(� of the area under the resonant peak of the product of SFz�f

�j � and jH�f�j �j��

The background response is the part of the dynamic response that is acting quasistatically due to slow variations of wind speeds� It covers a broad frequency bandbelow the lowest natural frequency� The resonant response is concentrated in apeak at the natural frequency� the height of which is controlled by the damping�For exible longspan structure� the resonant component dominates the response�especially in lift and torsion� For more rigid structure such as guyedmast towers�the background component generally dominates�

The contributions of the w component of the turbulence to the expressions for �B and �Rj

can be written as follows for the vertical forces�

�Bz�

�� V �BC �

z

�

�� w�V

�� Z �

�

f�Sw�f��

�wj Az�f

�� j� j Jz�f�� j� d ln f� ��

�Rzj�

�� V �BC �

z

�

�� w�V

�� f�Sw�f��

�wj Az�f

�� j� j Jz�f�j � j��

�sj ' �a�f�j ��

Similar expressions for the torsion can be written with � and m replacing z andintroducing an additional deck width B� For the lateral direction �Cx replaces C �

z

and u replaces w�If the left hand terms are normalized by the �� V �BC �

z�� term� the response is

a function primarily of the reduced frequency f� and the intensity of turbulence


� w� �V

�� two homologous quantities which link the fullscale bridge behaviour with

any dynamically scaled model� Otherwise� the turbulence controlled response isbound up in the functional form of the turbulence spectrum� Sw� the aerodynamicadmittance� Az� the joint acceptance function� Jz� and the aerodynamic damping�a�

Peak response� The peak response� �r� solution of the equations of motion� iscomposed of the following�

�r � �r ' g

r �B '

X �Rj

��

where �r is the mean response� g is a statistical peak factor� �B is the mean squarebackground response� and �Rj

is the mean square modal response at or near the

jth resonant frequency�

The peak factor g is based on the de�nition of the largest instantaneous value ofa stationary random variable with a Gaussian distribution�

g�T � �q� ln�T � '

��p� ln�T �

��

where is the cycling rate of the process and T is the sampling period� The longerthe sampling period� the smaller the variations of g� The cycling rate is de�ned by�

� �

R�

� f�SR�f�dfR�

� SR�f�df� ��

Typical values of g for the bueting response range from � to ��

Quasisteady aerodynamic damping� Linking damping with the velocity termof equation �� the vertical aerodynamic damping force can be expressed by�

Fz�damping � �B

��C �

z�V !z��

Damping as a fraction of critical is de�ned as the ratio of the work done by onecycle against the lift to the total energy stored in the system�

�aero�f�

j � �"E

��E�

G�j

Gcrit�

�B� �C �

z�V �

R l� �

�j�y�dy

�wjM�j

��B� �C �

z�V �

��fjm��

where G�j and M�

j are respectively the generalized damping and mass of mode j�and m is the mass per unit length of the deck �assumed here and in the followingas being constant along the span so that M�

j � mR l� �

�j �y�dy��


The expressions for the quasisteady aerodynamic damping as a fraction of critical�a�x�z�� and the aerodynamic stiness� �a�� reduce to�

�a�x �Cx

��

V

fxB

�B�

m$ �a�z �

C �z

��

V

fzB

�B�

m��

�a�� nC �

m

��

V

f�B

�B�

I�$ �a�� C �

m

��

�V

f�B

�� B�

I��

Aerodynamic damping from aerodynamic derivatives� The aerodynamicdamping of a given crosssection can also be obtained from wind tunnel experimentson section models or threedimensional aeroelastic models� in smooth or turbulent ow�

The estimation of the aerodynamic damping can be done in the time domainsimply by correlating the increase or reduction of the total damping of a mechanicalsystem with an increase in wind speed %�&$ or in the frequency domain by identifyingthe aerodynamic forces �� outofphase with the motion for various wind speeds%�&�

These experiments are generally linked with the determination of the aerodynamic derivatives that describe in detail the motioninduced forces� The ideal caseis to carry out a series of experiments that will make possible the system identi�cation in turbulent ow on the transient response as described by BogunovicJakobsen%��& or Zasso %�&� The identi�cation in smooth ow is however much simpler andreliable if motions are kept at small amplitudes %��&�

The motional aerodynamic derivatives can easily be converted to equivalent viscous damping and used directly in the calculation of the mechanical admittancefunction of the system or in equation �� The conversion of the aerodynamicderivative terms H�

� and A�� in Scanlan notation %�& can be made using the following

expressions %�&�

�a�z�f�

j � � �H�

� �f�

j ��B�

�m��

�a��f�

j � � �A�

��f�

j ��B�

�I��

�� Simpli�ed expression� resonant modal response

De�ning the resonant rootmeansquare modal response rRjby�

rRj�

q �Rj

K�j

where K�

j � �j

Z l

�m��j �y� dy� ��


and K�j and �

j are respectively the generalized stiness and angular velocity ofoscillation of mode j� Rearranging equation �� a general expression for thenormalized vertical resonant rms modal response �acceleration� can be obtained�

rjIwB�

j

�C �z

��

�V

fjB

��B�

m

sf�Sw�f��

�wjAz�f

��jjJz�f�j �js

��

�s ' �a�f��

Equation �� is a product of dimensionless parameters that can all be evaluatedwith a relatively high con�dence level through section model tests and full scalemeasurements at the site� It is valid for all modes where the deck dominates theaerodynamic forces and can be easily transformed to express the lateral and torsional responses� Note that if the structural damping is considered dominant inequation �� thus neglecting the aerodynamic damping term� the resonant rmsresponse will tend to be proportional to the reduced velocity at a power in thevincinity of � since Az�f

�� and Jz�f�j � are proportional to the reduced velocity %��&�

On the other hand� if the aerodynamic damping is dominant� the exponent willhave a value approaching � given the variation of �a with the reduced velocity inthe denominator of equation ��

�� Shortcomings

The frequency domain approach described in this section has been shown to be arelatively simple and reliable method to calculate the wind loads on a bridge deckdue to the bueting action of the wind� Its error margin is directly related to theextent of the knowledge of the aerodynamic properties �steady and unsteady� ofthe bridge deck considered� This is probably its main shortcoming�

If only the static coe�cients are known� error margins of the order of ��(overestimation of the rms response compared to measured response in full scaleor model scale can be expected %�&� If the unsteady properties are known� e�g�aerodynamic derivatives� aerodynamic admittance and spanwise coherence of theforces� this error margin can be considerably reduced to a more acceptable level�� '�� (�

Its main limitations are assumptions of linearity� the disregarding of couplingeects and the use of the strip assumption to de�ne the spanwise coherence of thewind loading� The latter is investigated in detail in Part II� It is reported in PartIIA that the error margin associated with the strip assumption only� varied froman overestimation of more than ��( to an underestimation of the wind loading of��(� depending on the reduced frequency when compared to direct measurementson section models�

Coupling� Coupling between modes is disregarded in the spectral approach described here� The error margin associated with this simpli�cation is hard to evaluatein a general manner�


Experience has shown that for service limit state calculations� where the windspeeds of interest are far from the range of aerodynamic instability� coupling eectsare small and are most often bene�cial� Coupling between modes will disorganizethe motion and transfer energy from one mode to an other which will appear asdamping�

In the early phases of the design� the structural engineer will aim at avoidingstructural coupling between the fundamental modes� In some instances however� acertain level of coupling is unavoidable� for example between a torsional mode anda lateral mode of an arch shape structure�

Linearity� The bueting theory is valid for bridge decks with static coe�cientsthat vary linearly with angle of wind incidence between say �� and '�� Herealso the designer will aim at choosing a crosssectional shape that satis�es thisassumption� For more blu sections or due to the presence of appendages such aswind breaks or crash barriers� this might not be possible�

For nonlinear variations of the static coe�cients� the error margin of the spectralapproach could be important and another method of evaluation is advisable� eitherexperimentally with an aeroelastic simulation or in the time domain where thenonlinear variations of the coe�cients can be modelled easily�

For what concerns nonlinear variations of damping or stiness of the structurewith amplitude of motions� the spectral approach by itself is not adequate� Damping for example has to be evaluated a priori for an assumed amplitude� This is oftenmade for conditions in the neighbourhood of the design wind speed� The combination of time domain bueting simulations and nonlinear �nite element analysisprogrammes can better deal with such eventualities� especially when ultimate limitstate conditions have to be studied %��&� A review of the developments of the timedomain bueting analysis is presented in the Section ��

Aerodynamic damping� For the vertical and the lateral degrees of freedom�quasisteady aerodynamics can in general predict the level of aerodynamic dampingof a deck crosssection within an acceptable error margin %�&�

However� for the torsional degree of freedom� quasisteady aerodynamics is oftennot adequate to predict the aerodynamic damping� In many occasions� quasisteadyaerodynamics using n � �� predicted positive damping while in reality� negativedamping �which adds energy to the motion� was present %�&� Experiments on sectionmodels to evaluate the torsional aerodynamic damping or the A�

��f�j � aerodynamic

derivatives are strongly recommended�

�� Improvements

Part IIA and Part IIB of this thesis deal with improvement of the current bueting analysis methods where it is believed that a substantial reduction of the error


margin of the predictions can be achieved� It concerns the application of the stripassumption and presents a model that departs from this assumption to calculatethe aerodynamic admittance and the spanwise coherence of the forces�

For nonstreamlined bridge decks� an experimental method to evaluate the aerodynamic admittance is proposed in Part IIIA� The method links crosssection admittance to the spanwise coherence of the forces on a span� The measured quantitycan be used directly in the current spectral approach to predict wind loading andbueting response� Other areas of improvement are discussed in the following�

Variations of the angle of wind incidence� To reduce the statistical uncertainties related to the linearisation of the static coe�cients used in the buetinganalysis� Irwin and Xie %��& have proposed an equivalent linearisation method�

It consists of calculating the weighted average values of the static coe�cientsusing a Gaussian probability density distribution for the instantaneous angle ofattack ��

�Cz�Iw� �

Z�

��

Cz�� p�� Iw� d�$ �C �z�Iw� �

Z�

��

dCz��

d�p�� Iw� d� ��

where

p�� Iw� ��p��Iw

exp

��

�

��

I�w

��

is a Gaussian distribution and Iw is the vertical turbulence intensity�This weighted average method is a clever way to deal with the problem of the

nonlinearity of the static coe�cients and can be introduced easily in the currentbueting theory� An example of the application of this method is given in Fig� �for static coe�cients measured in turbulent ow on a section model of a closedboxgirder bridge deck�

Calculation of the joint acceptance function� Under the strip assumptionthe root coherence of the forces can be represented by the root coherence of thewind velocity� In turn� the root coherence of the wind velocity is often expressedby an exponential function�

coh��L �"y� f�� coh��w �"y� f�� exp

��cf"y

�V

��

where c represents the width of the correlation and varies from � to �� dependingon the site �� is most often used� %�&�

Equation �� is an empirical expression and was found to be most valid forseparation "y somewhat smaller than the scale of turbulence� Collapse of the rootcoherence curves is expected for the smaller separation when plotted against reduced


−10 −8 −6 −4 −2 0 2 4 6 8 10−1.5

−1

−0.5

0

0.5

1

alpha (deg.)

Cz

a)

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.002

0.004

0.006

0.008

0.01

alpha (deg.)

p (a

lpha

)

I_w=0.06

I_w=0.12

b)

0 0.05 0.1 0.15 0.2

−0.044

−0.042

−0.04

I_w

Cz_

bar

c)

0 0.05 0.1 0.15 0.24

4.1

4.2

4.3

4.4

I_w

Cz_

prim

e_ba

rd)

Figure �� Application of the weighted average method for a closedbox girder bridgedeck� a� shows a linearization ��'�� thick line� of the measured coe�cients$b� shows the Gaussian probability of � for two levels of Iw$ c� and d� give respectively the variations of �Cz and �C �

z as a function of Iw using ��

frequency f"y� �V � which is the ratio of the separation to the wave length of the uctuations� For the case where the separation is of the order of or larger than theturbulence scales� �� is not adequate� the root coherence curves decaying like aseries of diminishing humps %�&�

Irwin %�& has shown that the use of this simple exponential expression couldintroduce an important overestimation of the response at low frequencies since itrepresents turbulence with in�nite length scales� In reality� the turbulence lengthscales are �nite and the root coherence for low frequencies and large separationsdrops considerably below �� where equation �� will put it� This was con�rmedon several occasions by other researchers� in model scale %�� & and in fullscale%��&�

Among others� Irwin suggested the use of an expression derived from the isotropicturbulence model based on the von K�arm�an spectrum� Part IIA of this thesisdescribes in detail the use of such a model to �t the root coherence data� It suggestsa correction for low frequencies and large separations of the parameter f"y�Bnormally used in the bueting analysis� The correction is based on turbulence


0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1C

o−co

here

nce

f delta_y / B

c=5.3

a)0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

von Karman parameter with Lw= 0.27

c1=0.73, c2=1.03, c3=0.27

b)

10−2 10−1 10010−2

10−1

100

Join

t acc

epta

nce

func

tion

fB/V

c)

Figure �� Variations of the joint acceptance function with reduced velocity for auniform mode shape ��y� � � for two representations of the wind cocoherence�a� solid line� cocohw�"y� f� � exp%�cf"y�B&$ b� solid line� cocohw�"y� f� �exp%�c��c� & cos�c�� On c� j Jz�f�j � j� was calculated using the �ts based on f"y�B�dashed line� and based on the von K�arm�an parameter �solid line��

length scales and is of the form�

� � k�"y

s� '

�

k��L�

��

where k� � ��f� �V � L is a length scale and � is called the von K�arm�an collapsingparameter %�&�

Fig� � compares curves of spanwise cocoherence of the wind velocity as a functionof f"y�V and �� and illustrates the eect of the two expressions on the calculation ofthe joint acceptance function� Cocoherence �or normalized cospectrum� is the realpart of the crossspectrum normalized by the onepoint spectrum and it is equivalentto the root coherence for isotropic turbulence� The error margin �overestimation�of the joint acceptance function associated with the simple exponential expressionat low frequencies is considerable and could be easily avoided�


Directionality� The problems associated with directionality of the incident windhave been dealt in detail in a modied bu�eting theory for yawed wind by Kimuraand Tanaka %��&� Directionality eects are often more important for bridges duringconstruction than in service conditions and should certainly be combined with theannual probability distribution of wind direction for a given site�

� Predictions in the time domain� a review

A great deal of work on the formulation of the unsteady aerodynamic forces ona thin airfoil in sinusoidal gusts and later in random gusts has been publishedsince the ��s %��&� The original formulations proposed by the pioneers such asSears� Wagner and Kussner have been further developed� especially in the ��sand ��s to a high degree of sophistication and are commonly used today in theaerospace industry�

At �rst� the formulations were developed in the time domain� then linearized� andthe problems were solved in the frequency domain %�&� It appears that this practice isstill true when it comes to general bueting problems of aircraft ying in turbulenceor bridges bueting in natural wind %��&� However� the development of computeraided ying and the active utter control of wings has pushed the need to expressthe unsteady aerodynamic forces in a time variant manner� Very e�cient algorithmswere developed in the time domain to perform the servocontrol of aps and arebased on aerodynamic data obtained experimentally in the frequency domain�

In the bridge aerodynamics �eld� the frequency domain approach is also predominantly applied to solve the problem of wind eects� even if the original formulationswere done in the time domain %�&� It was in the early ��s that Scanlan� Beliveau and Budlong �rst worked on solving the problem entirely in the time domain�introducing the indicial functions put forward earlier in the aeronautical �eld� Itis� however� only recently that much eort has been invested in developing e�cient time domain formulations of the unsteady aerodynamic forces that could becombined with a �nite element model of the structure and could include all thenonlinearities that have been omitted in the past %��&� This development has beenjusti�ed with the planning of very long bridges such as the bridge over the Strettodi Messina and the Akashi Kaikyo Bridge�

The following are brief remarks concerning what can be found in the recentliterature on this topic� It should be noted �rst that no less than �ve papers outof twenty on bridge aerodynamics were concerned with time domain predictions ofthe response at the last International Conference on Wind Engineering in January�� These papers give a good account of the stateoftheart in the formulationof the unsteady aerodynamic forces on bridge decks�

The problem of the formulation of the time varying wind load in the time domaincan be summarized as follows�


�� The bridge deck is excited by the turbulence of the wind at a certain time t

but also by what happened between the eddies in the ow and the deck at timet�z� That is to say that the wind loading has a memory or that there exists aphase lag and time lag between the source of the excitations and the loading oraerodynamic forces themselves� In the frequency domain this memory eectis expressed by an aerodynamic admittance function� The memory eect is asole function of the crosssectional geometry of the deck�

�� The motion of a deck in a moving uid induces forces that interact with theinitial forcing and creates new oscillations� These forces are called motioninduced forces or selfexcited forces and can be concentrated in one or severaldegrees of freedom �coupling of modes�� They are normally best described inthe frequency domain and their conversion to the time domain is problematic�

�� The aerodynamic forces are far from being fully correlated along the span ofthe bridge� The spanwise correlation of the approaching wind uctuations�combined with the mode shapes of the deck govern the spanwise distributionof the aerodynamic forcing� Also� the spanwise coherence of the aerodynamicforces is not necessarily identical to the spanwise coherence of the approaching ow� as stated by the strip assumption� None of the papers reviewed here havedealt with this aspect�

�� The simulation of the approaching wind has also its share of di�culties� Anadequate simulation should model all the onepoint and twopoint statistics ofthe wind uctuations� Locally it should respect the �rst and second momentsfor each wind component as well as auto spectrum and cross spectrum betweenthe components and spanwise� It should also model the cross spectra of therespective components� u� v and w� A discussion of this aspect is given in%��& and no comments will be made here concerning the approaches found inthe literature in connection with the wind loading formulation� Needless tosay� this aspect is of prime importance because� no matter how good the windloading algorithm is� if the input is not adequate� the output will suer�

In some instances it was found that the wind simulation was �trimmed� or��ltered� to account for some of the problems described above �e�g� aerodynamic admittance��

The approaches found in the literature can be classi�ed in �ve categories� i� quasisteady aerodynamics$ ii� corrected quasisteady aerodynamics$ iii� indicial functions�Fourier Transform�$ iv� rational function approximations �Laplace Transform�$ v�equivalent oscillator �neural network and black box��


�� Quasi�steady aerodynamics

Miyata et al� %��& clearly present the advantages of the time domain predictions ofthe action of wind on long span bridges when combined with a �nite element modelof the structure� The formulation used here is fairly simple and very much alongthe same lines as the original formulation by Davenport� Quasisteady aerodynamics is assumed as well as the strip assumption� Kovacs et al� %��& has a similarapproach except that the �nite element model of the structure includes structuralnonlinearities for large de ection in order to evaluate the ultimate state conditions�

Even though the limitations of these quasisteady aerodynamics approaches arenumerous� Miyata et al� %��& obtained surprisingly good agreement between simulation and wind tunnel tests for a very large suspension bridge �Akashi Kaikyo�� Thelimitations are� noninclusion of the motioninduced forces �besides quasisteadyaerodynamic damping� in the wind load model$ memory function expressed as anaerodynamic admittance �ltering the wind �eld simulation$ inadequate representation of the spanwise coherence of the forces$ and� the assumption that the centerof the aerodynamic forces is �xed at a distance �)� of the deck width from the deckcenterline� This assumption allows the inclusion of the torsional motion in the de�nition of the apparent angle of attack or in the relative wind velocity� The positionof the center of the aerodynamic forces is known to vary with reduced velocity�

�� Corrected quasi�steady aerodynamics

The work by Diana et al� %�� & in relation to the studies for the proposed bridgeover Stretto di Messina has taken a similar approach with the exception that someof the above limitations have been lifted by the development of a �corrected� quasisteady theory� Here� the motioninduced forces are fully included in the formulationof the aerodynamic forces via the experimentally determined motional aerodynamicderivatives� The location of the center of the aerodynamic forces is also determinedfrom the aerodynamic derivatives�

To include the motioninduced forces� Diana %��& proposed an equivalent linearisation� for each reduced velocity� of the type�

C�

L�� CLs�� '

Z �

��K�

L d�z ��

where�

C�

L�� corrected aerodynamic coe�cient$

CLs�� static coe�cient at angle of attack �� of the equilibrium position$

K�

L � aerodynamic derivatives �e�g� lift slope varying with reduced velocity��

Here � is de�ned as the apparent angle of attack based on the relationship�

� � � � !z�V� b�z

B !��V

'w�V

��


where z and � represent respectively a vertical displacement and rotation of thedeck� and b�z is the position of the point of applications of the vertical aerodynamicforces and is a function of reduced velocity�

This formulation has been shown to be adequate in many ways %��& except whenit deals with the response of the higher modes� The limitations attributed to thememory eects and the spanwise coherence of the forces are also not dealt with byDiana�

�� Indicial functions

In a series of papers spanning several years� Scanlan� Beliveau and Budlong %�� & described the use of indicial functions for the inclusion of the motioninducedforces in the time domain formulation of the aerodynamic forces� The formulationof the selfexcited lift forces Lse can be read as follows�

Lse ��

��V �B

dCL

d�

Z s

�%*L�s� �� ' +L�s� ��

z��

B&d� ��

where s � V t�B� �� d�ds and�

*L�s� � indicial response function due to a step change in angle of attack$

+L�s� � indicial response function due to a step change in vertical velocity�

*L�s� and +L�s� could be of the form

*L�s� � c� ' c�ec�s ' c�e

c�s ��

where the coe�cients ci of the indicial functions are extracted by nonlinear leastsquare �tting of the aerodynamic derivative curves� Even though this approachappears to be e�cient� very few examples of its application can be found in theliterature�

Recently� Lin et al� %��& proposed another model for the selfexcited forces on abridge deck based on the indicial function idea� The selfexcited loads are expressedin terms of convolution integrals of response functions due to a unit impulse displacement �vertical or angular�� These impulse response functions are obtained frominverse Fourier transformations of the frequency response functions of the form �formoments due to the change of angle of attack ��

FM�� B�V ��c� '

B

Vic� '

c�i

c�V�B ' i'

c�i

c�V�B ' i

��

FM�� B�V ��A�

� ' iA�

��

where the coe�cients c� to c� are obtained from nonlinear �tting of the aerodynamic derivative curves �A�

i and H�i � and K is a reduced frequency�


Li et al� %��& have presented a method to determine experimentally the coe�cientsc� to c� and therefore the impulse response functions� A complete example ofthe use of this formulation is given by Xiang et al� %��& for the motioninducedforces� Comparisons are made between time domain predictions and wind tunneltest results for the Shanton Bay Bridge� The agreement is remarkable� even thoughhere also the bueting forces are calculated using quasisteady aerodynamics�

� Rational function approximations

Fujino et al� %��&� in dealing with the problem of active control of utter for bridges�have formulated a method to express the motioninduced forces in the time domain�This method is directly inspired from work in the aerospace industry� Rationalfunction approximations are used to express the equation of motion of the deck ina linear time invariant statespace form where the unsteady aerodynamic forces donot depend explicitly on reduced velocity %�� &�

The aerodynamic derivatives obtained from experiments are stored in tabularform in the reduced frequency domain and are approximated in the Laplace domainby rational functions and a series of coe�cients� The rational functions are theninverse Laplace transformed to the time domain to solve the spacestate equation ofmotion �see Tiany and Adams �� %��&�� The level of accuracy is a function of thenumber of �aerodynamic states� used for the approximation� but once the numericalproblems are solved this method appears to be eective %��& and is currently usedin aerospace applications %��&�

Li and He %��& present a formulation of the unsteady forces that includes rationalfunction approximations for the motioninduced forces and the bueting forces areexpressed by convolution integrals of impulse aerodynamic transfer functions determined experimentally from wind tunnel tests� An example of the calculations isgiven but no comparisons with measured aeroelastic responses on a physical modelare made� This method appears to be the most complete of all the abovedescribedapproaches�

� Equivalent oscillator

Diana et al� %��& have initiated research on new methods to express the aerodynamic forces in the time domain including bueting� motioninduced forces andvortexshedding forces using numerical models comprised of the bridge deck andan equivalent oscillator� The new methods include �blackbox models�� a neuralnetwork model and sophisticated parameter identi�cation algorithms using an Extended Kalman Filter� Preliminary evaluations of the methods have shown satisfactory results� especially for nonlinear phenomena such as vortexshedding inducedoscillations�


Experimental approaches

The study of wind eects on long span bridges is possible to some extent throughtheoretical predictions but is almost always dependent on aerodynamic informationprovided by wind tunnel tests %��&� For example� the somewhat straightforwardprediction of the mean lateral de ection of a suspension bridge deck is dependenton the steady drag force coe�cient obtained from wind tunnel tests on a sectionmodel�

Wind tunnel testing of bridges involves the interaction of a structure with anoncoming ow� It is then important to stress that not only must the structure bemodelled properly but also the ow must be at the right scale in relation to thestructure and� moreover� representative of the wind at the prototype bridge site�

�� Section model method

Section model tests on a representative spanwise portion of a bridge deck is themost employed technique to obtain �rst level aerodynamic information on the deckcrosssection� Section models are used to detect signs of vortexinduced oscillationsand instability� as well as a source of a panoply of aerodynamic properties rangingfrom static force coe�cients to aerodynamic derivatives� They are limited by theirtwodimensional character but are often used early in the bridge design process tohelp de�ne the deck geometry�

The theoretical basis for extended section model testing in turbulent ow stipulates that for the section model and full bridge responses to be similar the followingrestrictions must hold %��&�

� the motioninduced aerodynamic forces at all locations across the span musthave the same linear function as the local motion of the deck �rigid model�$

� the scale of turbulence at the resonant frequency is small in comparison withthe length of the model �hence the term �extended� section model�� but stillin proportion with the geometrical scale of the model$ and�

� the aerodynamic forces on the cables and towers must be small compared tothose of the deck�

Early section model tests have shown satisfactory qualitative agreement with fullbridge model tests and have adequately predicted unstable behaviour of fullscalebridges %�� &� However� signi�cant de�ciencies in the earlier approach were noted%��& and an alternative approach was proposed by Davenport %��&� This approachincludes tests in turbulent ow as well as in smooth ow and utilizes the measuredresponse of an extended section model with suitable corrections for discrepancies inthe intensity and spectrum of turbulence� the damping and the mode shape� Theresponses are determined from estimates of the dynamic wind loads in the lowest


symmetric and asymmetric modes as well as the steady wind loads for the lateral�vertical and torsional degrees of freedom�

From these experiments it is possible to extract the aerodynamic input necessaryfor the application of the bueting theory and predict analytically the responseof the prototype bridge� However it is also possible to convert the section modelresults to fullscale equivalent static loads� as described below� and to apply thisloading to the prototype structure to calculate the bueting response�

�� Predictions of full�scale response

Static response� The static part of the response �or the mean response� canbe predicted from the static force coe�cients of the deck obtained from the sectionmodel tests� The lateral and vertical steady wind forces applied to the deck throughthe cables can be signi�cant and should be included in many cases %��&� The eectof the mean lateral displacement of the top of the pylons should also be includedin the predictions of the mean response�

The static coe�cients� determined in turbulent ow and expressed in the decklocal coordinate system� are converted to fullscale static wind forces as a functionof the wind speed� The static response is obtained by dividing the static wind forcesby the generalized stiness of the fundamental mode studied�

For example� the mean lateral amplitude "�x of the deck can be estimated fromthe following expression if the mean drag coe�cient Cx� evaluated at �� of windincidence� and the fundamental horizontal mode shape ��y� �normalized to unitmaximum amplitude� are known�

"�x

B�

�FxBK�

x

��

�V �CxlB

B�x

R l� m�y��y��dy

� ��

The mean horizontal displacement at a given point y on the deck is then givenby "�x��y� � �x�y� so that �

�x�y�

B�

��

�V �Cxl

��fx��mR l� �

��y�dy��y��

assuming a constant mass per unit length m�y� � m�

Dynamic response� The background and resonant components of the dynamicresponse are expressed respectively by equations �� and �� These expressionsapply to both the full scale bridge and a section model in a turbulent wind tunnel ow� If the main characteristics of the full scale turbulence can be reproducedadequately in the wind tunnel with respect to the scale of the model� the resultsof the dynamic tests can be directly adapted to full scale with minimal corrections�Tests of extended section models �with �� aspect ratio or the like� in a turbulent


ow generated behind large spires have shown that it is possible� Although theturbulence scale and intensity cannot be made to correspond exactly to full scale�the values can be made su�ciently close�

To adapt the results of the dynamic section model testing to full scale� correctionsneed to be applied as follows�

�� Corrections of the low frequency background response� �B � largely omittedfrom the section model due to the de�cit in the vertical velocity spectrumgenerated by the spires� This can be estimated with reasonable accuracyfrom equation �� using either theoretical or experimental data as outlinedbelow�

�� Corrections of the terms in the resonant response �R for the discrepancies inthe turbulence intensity� vertical velocity spectrum� joint acceptance functionand damping�

The correction factors are found from the ratios of the quantities involved formodel and fullscale� For example� the correction factor for the turbulenceintensity is a constant �i�e� the ratio of target full scale to model values��Similarly� the correction factor for the vertical velocity spectrum is the ratiobetween the target full scale spectra and the model scale spectra at a given frequency� The joint acceptance function correction� converts the uniform modeshape values of the section model to the quasisinusoidal mode shapes of theprototype� The damping correction combines structural damping and aerodynamic damping to satisfy equation �� Through the appropriate selectionin the section model tests of turbulence characteristics� structural dampingand geometrical scale� these corrections can be kept relatively small�

Comparisons� For an adequate wind simulation� the extended section modelmethod is superior to the analytical spectral approach in many ways� None of thelimiting assumptions of the bueting theory are necessary and it can also simulatemotioninduced forces in turbulent ow or with nonlinear behaviour� It has thedisadvantage that a new series of bueting tests is required for each additionalstructural con�guration� e�g� frequency ratio or generalized mass�

�� Three�dimensional physical models

Fullbridge aeroelastic model� Wind tunnel testing of full bridge aeroelasticmodels in adequately simulated atmospheric boundary layer ow is the best methodof prediction of the response� besides testing the prototype bridge itself in naturalwind� It allows the determination of the stability of the bridge in high winds�investigation of any vortexinduced oscillations and characterization of the buetingresponse %��&�


The main character of this technique is its three dimensionality� Threedimensionalwind gusts exciting �D pylons� cables and deck with �D vibration mode shapes isprovides a wealth of information for bridge aerodynamicists� Nonlinearity eects�mass and stiness distribution eects� nonuniform de ected shape eects as wellas aerodynamic coupling eects are included in the simulation�

This approach is most suited for the study of the behaviour of the structure incomplex terrain� for yaw winds� in service and for some critical construction stages%��&�

Complexity of the model design and construction� as well as higher costs limit thistechnique� Froude number similitude is generally respected �a must for suspensionbridges where the geometrical stiness of the main cables is the dominant stiness��which suggests that the bigger the model the better if the aerodynamics of thestructure at low wind speeds should be studied� However� limitations in the sizeof the experimental facilities for testing models of very long span bridges at largescale seems not to be a factor any longer with the construction of very large windtunnels to accommodate such models�

Taut strip model approach� This approach oers most of the advantages ofthe fullbridge aeroelastic model study at a lower cost� The pylons� the cables andthe side spans �if any� are not modelled but the �D character of the motion of themain span is respected� Froude number similitude can be relaxed which can be animportant advantage when both vortexshedding induced oscillations at low windspeeds and aerodynamic instability at high wind speed should be studied�

For this approach� the length scale and the time scale can be selected independently� It can be described as an extended section model test where many �Dmodes are studied at the same time� Coupling eects between the vertical andtorsional modes are modelled� however� only for the �rst fundamental symmetricmodes� A thorough description and analysis of this approach is given in %�&�

Concluding remark� The treatment of the three dimensionality of the windloading on linelike structures by the analytical prediction methods is the sourceof an important level of uncertainty� Therefore� the experimental methods representing the �D character of the windstructure interactions continue to be superiorto the theoretical approaches� Part IB focuses on the description of the spatialdistribution of the wind loading due to wind gusts�

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%��& Davenport A�G�� Isyumov N�� Fader D�J� and Bowen C�F�P�� A study of wind ac�

tion on a suspension bridge during erection and completion� University of Western Ontario� Canada� Research Report BLWT�� May �� and March��

%��& Larose G�L� and Livesey F�M�� On cablestayed bridge during construction�modelling and prediction of aerodynamic behaviour�� in Proc� of AFPC Conf�

Cable�stayed and Suspension Bridges� Vol�� Deauville� France� Oct� ��

Part I�B Span�wise coherence of wind forces ��

Part I

Background

Part I�B

The span�wise coherence of wind forces



mark� �� Lyngby� Denmarkb Danish Maritime Institute� Hjortek�rsvej � �� Lyngby� Denmark

Abstract

Forces due to gusts acting on a streamlined bridge deck can be more correlated alongthe bridge span than the oncoming wind velocity uctuations� This implies nonnegligible secondary cross ows and brings some limitations to the strip assumption�

Based on a review of the eects of turbulence on the aerodynamics of a blubody� a description of the ow mechanisms that could explain the observed largercorrelation of the forces is proposed in this paper� A summary of the empiricalexpressions recently proposed to model the spanwise coherence of the forces onlinelike structures in turbulent ow is also given�

� Joint acceptance function and span�wise coherence

The joint acceptance function is de�ned in Part IA �Section �� as a measureof the correlation between the spatial distribution for the bueting forces across aspan and the mode shapes of that span� It allows the calculation of the wind loadapplied on a span given the wind load on a chordwise strip�

The problematic part is to de�ne the spatial distribution of the forces given anincident turbulent ow �eld� This question is at the heart of all prediction methodsof the response of linelike structures to turbulent wind� either analytically in thetime domain and the frequency domain or experimentally�

�Parts of this paper have been published in Proceedings of the Third International Colloquium

on Blu� Body Aerodynamics and Applications� Blacksburg� Virginia� July ��


Given the forces F� and F� acting on a bridge deck at the spanwise positionsy� and y� at time t and causing the excitation of the jth mode shape �j� the meansquare of the time varying wind force on the span with length l can be expressedby %�&�

F �j �t�"y� �

Z l

�

Z l

�F��t� y��F��t� y�� j�y��j�y�� dy�dy� ��

where the term F��t� y��F��t� y�� represents the covariance of the bueting windloads between point y� and y� on the span�

In the frequency domain� the covariance is transformed into a crossspectrumwhich can be normalized by the onepoint spectrum of the forces on a strip� SL� toobtain a normalized crossspectrum of the load� RL�f�"y�� Equation �� becomes�

SF �fj� � SL�f�

Z l

�

Z l

�RL�f�"y��j�y��j�y��dy�dy� ��

where the double integral is known as the joint acceptance function� j J�fj� j��The spatial distribution of the forces is represented by RL�f�"y�� As a starting

point and for many cases it is reasonable to assume that the spatial distribution ofthe incident wind will be representative of RL�f�"y��

On the basis of the strip assumption and for cases where the size of the structureis small in comparisons to the size of the gusts� Davenport proposed %�&�

RF �f�"y� � Rw�f�"y� � coh��w �f�"y� ��

where w is a component of the wind uctuations and coh��w is the spanwise root

coherence of w�

The magnitude squared coherence function is de�ned as the ratio of the squareof the real and imaginary parts of the crossspectrum between two points� to theproduct of the autospectra at these points�

coh�f�"y� �j Co��f� 'Qu��f� j�

SL��f�SL��f��

where� Co��f� � Cospectrum� i�e� real component of the cross

spectrum between points � and �$

Qu��f� � Quadrature spectrum� i�e� imaginary �outofphase�

part of the crossspectrum�

The root coherence is the square root of equation �� For the bueting problem�the contribution of the quadrature spectrum is nil even though Qu��f� of the wind


can represent a nonnegligible phase angle� The root coherence is thus replaced bythe cocoherence or normalized cospectrum which is de�ned as�

cocoh�f�"y� �Co��f�p

SL��f�SL��f��

There is evidence now that the limitation of the strip assumption is an importantsource of uncertainty� especially for structures such as closedbox girder bridge deckswhere it is simply not valid� The spanwise coherence of the aerodynamic forceson a bridge deck was found to be larger than the coherence of the oncoming ow uctuations�

The main objective of this paper is to describe the ow behaviour when it ispassing by a twodimensional blu body that has crosssectional dimensions of similar size to the size of the gusts� A second objective is to give a summary of theempirical models recently proposed to express the spanwise coherence of the forcesfor bridge decks in order to introduce the experiments reported in Part IIA�

� Flow over a bridge deck with sharp edges

To determine the cause of the observed larger spanwise coherence of the forcesthan the incident ow it is important to begin with the description of the mainfeature of the ow around a sharpedged body� The dominating phenomenon isthe generation of a shear layer at the leading edge �at a sharp corner�� The shearlayer might or might not reattach depending on the length and geometry of theafterbody� the turbulence of the ow� etc� In the shear layer� a recirculating zoneis formed de�ning a separation bubble�

Experience has shown that it is in the separation bubble that the largest peakpressures� thus forces� are formed on the body %�&� For closedbox girder bridgedecks� separation bubbles on the top and bottom surfaces are formed and dominatethe surface pressure distribution %�&� Referring to Fig� �� the following is a summaryof observations extracted from studies of the reattaching shear layer for sharpedgedbodies�

Smooth ow� chordwise and spanwise�

� Rolledup vortices are initiated at the separation point� a separation bubbleis formed and in the bubble %�� &�

� smaller eddies are stretched to larger scales and contact with the surfaceat x�xR � �� where xR is the chordwise location of the reattachmentpoint�$

� in the center of the bubble� the largescale coherent vortices are contaminated by smaller eddies$


� larger scale vortices are found at the surface and outer layers�

� Weak production of eddies in the bubble up to a certain amount inducing agrowth of the bubble� followed by an explosion to shed a larger vortex� andthen contraction of the bubble�

� Vortices are shed at the reattachment point at fxRV � ��

� Flapping motion of the shear layer at fxRV � �� is initiated�

� Lxu is constant� � ��xR� across the bubble due to contamination of small

scale eddies�

� Lyu is equal to �� Lx

u at the center of the bubble� but Lyu �� Lx

u at the edgeof the shear layer�

� Amalgamation of rolledup vortices produces larger and larger vortices� thisprocess is faster spanwise than chordwise creating an increase of the spanwise correlation of velocity uctuations and surface pressures�

Turbulence e�ect� chordwise�

� Oncoming turbulence mixes up with rolledup vortices� increases entrainmentrate and reduces the bubble length %�� &� For example� xR � ��D insmooth ow and xR � ��D in turbulent ow %�&�

Figure �� Notation for the description of the ow over a sharpedged body� Thedashed line marks the boundary of the separation bubble�


� When Iu increases� xR decreases but the peak suction increases and maximumrms Cp moves upstream %�� &�

� When Lxu increases� Cp stays constant but rms Cp increases %��&� for example�

when Lxu increases by �� rms Cp increases �� %�&�

Turbulence e�ect� spanwise�

� When Iu increases� xR reduces� the bubble is more �D and RP�P� increases�spanwise correlation of surface pressures� and then reaches a plateau�

� Out of the reattaching layer� Lxu increases when Iu increases� and the diusion

of the shear layer vorticity is enhanced�

� When Lyu�D increases� RP�P� increases and the bubble is elongated� therefore

more �D %�� &�

E�ect of chordtothickness ratio�

� When B�D increases� RP�P� increases� the bubble size reduces in length andheight� and the rms Cp reduces which helps the bubble to become more �D%�&�

� Some e�ects of the deck on the turbulence

In light of the above� the following description of some eects of the deck on theoncoming turbulence are proposed to explain the larger spanwise correlation� Thebridge deck�

� creates a distortion of the turbulence� i�e� stretching and rotating of thevortex line �laments as they are convected past the bodies� The distortionfollows the mean velocity streamlines %�� &�

For the larger vortices� Lxw�B � �� and for larger B�D� the energy in w is

transferred to u and v �splashing� as they are passing over the deck�

For Lxu�D � �� u increases and v� w remain constant$

� breaks up some of the large vortices� The resulting smaller scale turbulencemixes up with the shear layer to increase the entrainment rate� The separationbubble reduces in length but becomes more elongated spanwise �more �D�$

� sheds bodyinduced vortices� These vortices stretch more rapidly spanwisethan chordwise�


The combination of the distortion created by the deck on the approaching turbulence� the physical averaging by the deck of the wind uctuations and the �Dcharacter of the separation bubbles can partly explain the observed larger correlation of the forces� It suggests also that the larger the chordtodepth ratio� thelarger the spanwise coherence of the wind forces�

� Empirical models of span�wise coherence

In this section� a review of the empirical models of the spanwise coherence of theforces due to turbulence on a motionless bridge deck is given�

�� Models based on the strip assumption

If the strip assumption holds� the spatial distribution of the wind velocity uctuations su�ces to describe the spanwise coherence of the forces�

Davenport proposed one of the �rst empirical models of the spatial distributionof the u component of the wind based on �eld measurements in the atmosphericboundary layer %��&�

coh��u �f�"y� � exp%�cf"y

V& ��

where c is a �tted constant de�ning the extent of the correlation and f"y�V is areduced frequency also used as a collapsing parameter�

This expression has become almost a standard to solve the bueting problem�Solari %��& has made an extensive review of all the dierent values of c found in theliterature for various sites and atmospheric conditions� Equation �� oers a simplesolution to a complex problem but is associated with a certain level of uncertaintyespecially at low frequencies and large separations �see Part IA� Section �� anddoes not always �t particularly well the observed data�

To improve the �t of fullscale data for a sea exposure� Naito %��& has added anexponent to �� so that�

coh��u �f�"y� � exp

��c� f"y

V

�c��

where c� varies between �� and �� depending on the site�

Another approach is to �t the spanwise root coherence data with an analyticalmodel of the crossspectrum calculated from a model of the onepoint spectrum�Typically� the isotropic turbulence model based on the von K�arm�an spectrum isused and it was found to �t the coherence data fairly well even for nonisotropicwind conditions�

As described in Part IA �Section �� Irwin %�& has used this approach� as wellas Thompson %�& and Roberts and Surry %�&� It has the advantage of �tting well


the data for the low frequency and large separation range where �� fails� A morecomplete model of the spatial distribution of an isotropic wind �eld based on thevon K�arm�an spectrum was derived by Mann et al� %��& and is used in Part IIA ofthis thesis�

�� When the strip assumption is not valid

To the author�s knowledge� Nettleton was the �rst to demonstrate the eect on thespanwise variations of the lift forces of having gusts passing by a thin airfoil witha larger chord than the vertical scale of the turbulence �reported in %�&�� For thiscase� the strip assumption is not valid and he observed that the forces were morecorrelated than the incident wind�

The width of the correlation can be represented by the integral length scale as�

Ly �

Z�

�R��"y� d�"y� ��

where R�� is the crosscorrelation coe�cient� It was found from Nettleton�s datathat the correlation width of the forces was �� times larger than the correlationwidth of the incident wind �Ly

w�� m and B�� m��

LyL � �� Ly

w� ��

Melbourne %�& came to similar conclusions by comparing the spanwise coherenceof the leading edge pressures on the West Gate Bridge to the coherence of theincident wind velocity� The experiments were �rst conducted in full scale on theprototype bridge and veri�ed later in model scale in grid turbulence� It is the onlyfullscale investigation of the spatial distribution of the wind pressure ever publishedfor a bridge deck�

Melbourne�s results clearly indicated a larger spanwise coherence of the pressures� Obviously it was not possible to restrain the prototype deck from movingwhich might have increased the coherence of the pressure� However� the results areconvincing� especially when compared to model scale�

Melbourne �tted the coherence � data with exponential functions� He obtained�

cohpressure � exp%��f"y

V& ��

for the fullscale and modelscale leading edge pressures �at ��( of B downstreamof the leading edge� and�

cohu � exp%��f"y

V& ��

for the incident velocity uctuations of the u component�

�Since reference �� does not give a denition of the coherence function used in the paper� it hasbeen interpreted as being the coherence as in �� and not the root coherence�


Empirical model by Hjorth�Hansen et al�

HjorthHansen et al� presented in %��& an empirical model of the root coherenceof the aerodynamic forces based on the exponential decay function �� to whicha correction term was added� The correction term takes into account the lack ofcoherence of the forces at low frequency and large separations� It includes directlythe integral length scale of the aerodynamic force� Ly

D� as a key parameter for thecorrection� HjorthHansen et al� remarked that the integral length scale of the dragforces was found to be from ��( to ��( larger than the integral length scale ofthe ucomponent of the wind� Ly

u�The coherence data was obtained from wind tunnel experiments on a bridge deck

with a rectangular crosssection �larger depth than width� with constant width butwith varying depth spanwise� The forces were obtained by integration of unsteadysurface pressures simultaneously on three strips�

The root coherence data was �tted with an exponential function of the form�

coh��f�"y� � exp

��

��c��f"y�V � ' c�

�"y

LyD

��

The values of the constants c� � �� c� � � � � and LyD � �� m were observed

as providing the best �t to the spanwise coherence of the drag force� The best �tfor the u component was c� � �� and c� � � and Ly

u � �� m�HjorthHansen et al� also noted lower than anticipated drag forces on a strip

when compared to quasisteady bueting theory and suggested that the observedlarger coherence could be compensated for by a lower crosssectional admittance%��&�

Empirical model by Larose

A study of the spatial variations of the lift and pitching forces on a closedboxgirder bridge deck in turbulent boundary layer ow was carried out by Larose andreported in %�� &� The section studied had a widthtodepth ratio of �� and hadthe same fairing proportions as the decks studied in the present research�

The unsteady forces were obtained by integration of surface pressures on chordwise strips of a �xed model� The spanwise correlation and root coherence of theforces were found to be considerably larger than the incident wind characteristics�even though the length scales of turbulence were adequately scaled in relation tothe size of the model to represent fullscale conditions� The ratio of the length scaleof the vertical gusts to the deck width was Lw�B � �� These experiments areunique since they were conducted in a properly simulated atmospheric boundarylayer ow �eld �nonisotropic��

The variations of the crosscorrelation coe�cient R�� with "y�B obtained byLarose are reproduced in Fig� � for the vertical and torsional aerodynamic forces and


0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

delta_y/B

Cro

ss−

corr

elat

ion

coef

ficie

nt

Lift Torsionu−comp.w−comp.

Figure �� Correlation of the aerodynamic forces and wind uctuations as a functionof normalized separation� after Larose %�&�

the longitudinal and vertical components of the boundary layer ow� By integrationof these curves the correlation width can be calculated as follows�

LyL � �� Ly

w� ��

An attempt was made by Larose to collapse the forces root coherence curveswith the von K�arm�an parameter that had modelled well the incident ow� Thecurves did not collapse but were however organized in a manner suggesting a "y�Bdependence� Larose thus proposed adding a reduced frequency term fB�V to thevon K�arm�an collapsing parameter so that�

�� k�"y

s� '

�

��k�Lw��'fB

V��

For very small B the added term is negligible and the root coherence of the forcesis then represented by the root coherence of the wind� A good collapse of the rootcoherence of the bulk of the data was obtained with �� and the resulting curve was�tted with an exponential function�

coh��L �k��"y� � expf�c��c�g � expf�c�

�� ' c�

fB

V

�c�

g ��


10−2 10−1 10010−2

10−1

100

Join

t acc

epta

nce

func

tion

fB/V

Figure �� Joint acceptance functions for a uniform mode shape ��y� � �� calculatedusing a �t of the wind velocity root coherence �dashed line� and using the measuredroot coherence of the lift forces �solid line�� after Larose %�&�

where� c� � �� c� � �� c� � � for the vertical wind uctuations$

c� � �� c� � �� c� � � for the vertical forces$

c� � �� c� � �� c� � � for the torsional forces�

Using �� and the coe�cients above� the joint acceptance function can be calculated using both the forces directly or the wind as representative of the spatialdistribution of the forces� Larose reported the calculations in %�&� A similar comparisons is shown in Fig� � for a span length of � m�

Empirical model by Bogunovic Jakobsen

Similar experiments were carried out by Bogunovic Jakobsen %�� & on a �� scalemodel of closedbox girder bridge deck with B�D� �� The tests were conductedin grid turbulence with Ly

w � �� m� on a motionless section model �� m longand �� m wide resulting in a ratio Ly

w�B � �� for these experiments�

The unsteady forces were calculated by integration of surface pressure measurements simultaneously on � chordwise strips of the model� Bogunovic Jakobsencompared the measured forces to the incident wind and obtained for the correlation


width�

LyL � �� Ly

w� ��

With a least square algorithm� Bogunovic Jakobsen �tted the cocoherence datawith the same exponential function for both the wind and the forces� The functionhas a builtin correction for the low frequency large separation range and is of theform�

coh��L�w�f�"y� � exp

��

�"y

qc�� ' �c�f�V ��

�c��

where c�� c� and c� are �tted to the observed cocoherence data�

The results of the �t are summarized in Table �� Using �� and the coe�cients ofTable �� Bogunovic Jakobsen calculated the joint acceptance functions and obtainedan important underestimation of j J�fjB�V � j� when the strip assumption wasassumed to be valid� The calculations are reproduced in Fig� � for a span length of�� m�

Table �� Least square �t of cocoherence data using �� after Bugonovic Jakobsen�

c� c� c��m��

Wind� w component �� Forces� Lift �� Forces� Torsion ��

Empirical model by Kimura et al�

Kimura et al� have reported in %�� & experiments on at cylinders in grid turbulence focusing on the description of the chordwise and spanwise distribution ofthe forces� Their main conclusion is in agreement with the results described above�with the exception that for the drag forces� the spatial distribution of the incidentwind described well the distribution of the forces %�&�

Three crosssections were studied� a at hexagonal cylinder with B�D � ��and two rectangular cylinders with B�D� �� and �� The width of the modelswas kept constant at �� m� The forces were obtained by integration of surfacepressures on chordwise strips of the models� The eects of the scales of the gusts onthe bueting forces was studied using three dierent grids resulting in the followingLyw�B ratios� �� and ��

The analysis of the incident wind �eld was based on the work by Irwin %�& whoderived analytically a root coherence function of the w component based on thevon K�arm�an spectrum� This function �tted well the wind root coherence data�Kimura et al� attempted to �t the lift force root coherence data with the same


expression� The �t was not satisfactory and to improve the �t� modi�cations tothe analytical expression were made� The measured turbulence length scale wasincreased by a factor of �� and the frequency was raised to a power of �� Themodi�ed von K�arm�an root coherence function proposed by Kimura et al� is of theform�

coh��L �f�"y� � ��

�B��K��

��K��

� ' ��hf��L

yw

V �i�

�CA ��

where

� �"y

��Lyw��

vuut� ' ��

f��

��Lyw

V�

��

K�� are modi�ed Bessel functions of the second kind and Lyw is a spanwise

length scale of the w component of the wind� The numbers in bold face representthe modi�cation made by Kimura et al�

This empirical expression �tted well the spanwise cocoherence curves of thelift forces for the three crosssections studied� Kimura et al� also compared themeasured lift spectrum on a strip to theoretical calculations based on the buet

10−2 10−1 10010−2

10−1

100

Join

t acc

epta

nce

func

tion

fB/V

Figure �� Joint acceptance functions for a uniform mode shape ��y� � �� calculatedusing a �t of the wind velocity cocoherence �dashed line� and using the measuredcocoherence of the lift forces �solid line�� after Bogunovic Jakobsen %��&�


ing theory� They observed a smaller crosssectional lift force than anticipated� inagreement with results reported by Larose in %�&�

Kimura et al� suggested that this discrepancy was due to threedimensional eects�or secondary cross ow� that reduces the force due to an incident gust chordwisebut increases its in uence spanwise� They also remarked that the smaller the scaleof turbulence the larger this eect�

Concluding remarks

The description of the behaviour of the vorticity in the reattaching shear layerpresented in Section � can partly explain the observed spanwise redistribution of thewind pressure when gusts are passing by a closedbox girder bridge deck� However�the ow mechanisms described here happen at high frequencies corresponding tothe frequency range of the bodyinduced turbulence� Experiments have shown thatlarger spanwise coherence was also observed in the low frequency range� Somethingother than that described here must happen to the larger incident gusts at lowfrequencies�

The spanwise coherence of the aerodynamic forces has been expressed by

coh��L �f�"y� � coh��w �f�"y� � expf�c�f"y

V�g ��

on the basis of the strip assumption� Several experimental studies have indicatedthat the above expression was not valid and was an important source of uncertaintyfor the calculation of the bueting response�

Empirical models have been proposed to represent better the spatial distributionof the forces on linelike structures� The models based on the von K�arm�an spectraltensor are the most promising since they represent well the incident wind conditionsand could be adapted to model the forces�

In general� the empirical models failed to describe the ow mechanisms involved�Intuitively� the root coherence of the forces should be a function of�

coh��L � F

�k��"y�

B

Lw�B

D

��

The experiments reported in Part IIA of this thesis aim at de�ning the relationship between crosssectional admittance� spanwise coherence and the in uence ofthe size of the gusts compared with the size of the bridge deck on these quantities�Another objective is to study the in uence of the deck aspect ratio B�D on theunsteady aerodynamic forces� It is believed that it could provide the missing information needed to build a more general model of the spatial distribution of the liftand pitching forces on closedbox girder bridge decks and similar blu bodies�


References

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%�& Saatho P�J� and Melbourne W�H�� The generation of peak pressures in separated)reattaching ow�� J� Wind Eng� Ind� Aerodyn��

%�& Larose G�L�� The response of a suspension bridge deck to turbulent wind� the

taut strip model approach� M� E� Sc� Thesis� The University of Western OntarioLondon� Ontario� Canada� March ��

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��

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%��& Davenport A�G�� The spectrum of horizontal gustiness near the ground inhigh winds�� J� Royal Meteorological Society� ��

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%��& Naito G�� The spatial structure of surface wind over the ocean�� J� Wind Eng�

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Near The Ground� Part III� variations in space and time for strong winds�Data Item �� October ��

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%��& Kimura K� and Fujino Y�� Turbulence scale eects on bueting forces of a at hexagonal section�� in Proc� of the rd Asia�Pacic Symposium on Wind

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Part II�A Measurements of bu�eting forces ��

Part II

Experiments and Analysis

Part II�A

Direct measurements of bu�eting wind forces

on bridge decks


a Danish Maritime Institute� Hjortek�rsvej � �� Lyngby� Denmarkb Department of Structural Engineering and Materials� Technical University of Den�

mark� �� Lyngby� Denmark

Abstract

Direct measurements of the forces induced by the bueting action of wind on amotionless bridge deck have been carried out at the Danish Maritime Institute�DMI�� The wind tunnel experiments aimed at de�ning the spatial distribution ofthe wind loading as a function of the deck width and the scales of the oncomingturbulence�

This paper presents the main �ndings of the experiments and illustrates theimpetus behind this research which was to evaluate� for closedbox girder bridgedecks� the error margin of the wind load predictions based on the strip assumption�

� Introduction

A research project aimed at de�ning the error margin of the bueting responseprediction for linelike structures highlighted a number of shortcomings of the current theory� These included� the linearisation of the wind loading� the evaluationof the aerodynamic damping and crosssectional admittance� and the theoretical

�A � �page summary of Part II�A has been published in J� of Wind Engineering and Indus�

trial Aerodynamics �� co�authored by Hiroshi Tanaka� Niels J� Gimsing andCla�es Dyrbye�


modelling of the transition from pointlike forces to dynamic loads on an entirespan�

The relationship between incident wind velocity uctuations and resulting windloads is an important source of uncertainty in bridge aerodynamics� To studythis aspect� a series of experiments were conducted with the objective of directlymeasuring the dynamic wind loading on motionless streamlined bridge decks inturbulent ow�

A second objective was to conduct a parametric study of the in uence of the deckwidth� railings� turbulence intensity and scales of turbulence on the bueting windloads� The purpose of the parametric study was to develop a threedimensionalmodel of the wind forces induced by turbulence on bridge decks�

This paper presents the main �ndings of the experiments as well as a succinctdescription of the methodology employed� The background for the research is givenin Part IB�

� Description of the experiments

�� Instantaneous force measurements

A �� m long section model was rigidly mounted in DMI�s Wind Tunnel II thathas a �� m wide � �� m high � and �� m long working section� Any windinducedmotions of the model were prevented using guy wires stretched between the modeland the wind tunnel boundaries� The eigen frequency of the structural system wasfound to be above �� Hz� The �� m long centre portion of the model was made ofacrylic and was instrumented to measure unsteady surface pressures� The rest ofthe model was made of a combination of wood and foam and was put in place onlyto ensure the �dimensionality of the ow and to avoid end eects� Photographsdescribing the experimental setup are presented in Appendix�

The dynamic force measurements were based on simultaneous measurements ofsurface pressures around three chordwise strips of the centre portion of the model�Each strip was �tted with at least �� pressure taps and the spanwise separationof the strips was adjustable� Three electronic pressure scanners �PSI ESP��HD�were mounted directly inside the model to keep the tubing length �� mm� to aminimum ensuring a good frequency response up to at least �� Hz�

The frequency response of the tubing system �hard vinyl tubes with � mm insidediameter� was veri�ed insitu by comparing the autospectra of the surface pressuremeasured on the windward face of a square plate placed normal to the oncomingturbulent ow for various tubing lengths� The smallest possible length of the vinyltubing �� mm� was used as a reference� Fig� � compares the frequency response ofthe reference length with the tubing length used in this study�

A similar veri�cation of the frequency response of the tubing was done in the laboratory� For this case� the source of uctuating pressures was a loudspeaker driven


0.5 1 5 10 20 40 8010−1

100

101

102

103

Frequency (Hz)

Spe

ctra

l den

sity

, S(f)

15 mm tube: +; 150 mm: ___

(a)

1 5 10 20 40 800

1

2

Mag

nitu

de

Transfer function

1 5 10 20 40 80

−50

0

50

Frequency (Hz)

Pha

se (d

eg.)

(b)

Figure �� Insitu frequency response of a typical tube used in this study$ a� comparisons of autospectra of surface pressures� b� variations of the magnitude andphase of the transfer function between the reference tube and a �� mm tube�

by a white noise signal� Spectral comparisons up to �� Hz between the reference�� mm tube and tubes of various lengths showed good behaviour in magnitude andin phase for tubing lengths up to �� mm� even though the uctuating pressuresfrom the loudspeaker had a resonant peak at �� Hz as seen in Fig� ��

The crosssection studied was typical of a closedbox girder bridge deck �Fig� ��The model was made in modules so the widthtodepth ratio� B�D� of the crosssection could be changed from � to �� and �nally �� the depth being keptconstant at �� mm �Fig� �� The edge con�guration �taken similar to the Storeb�alt

0.5 1 5 10 20 40 80100

101

102

Frequency (Hz)

Spe

ctra

l den

sity

, S(f)

15 mm tube: +; 150 mm: ___

(a)

1 5 10 20 40 800

1

2

Mag

nitu

de

Transfer function

1 5 10 20 40 80

−50

0

50

Frequency (Hz)

Pha

se (d

eg.)

(b)

Figure �� Tubing frequency response to uctuating pressures produced by a loudspeaker� a� comparisons of autospectra$ b� transfer function between the referencetube and a �� mm tube�


East Bridge con�guration� was kept constant� Note that the B�D ratio of �� onFig� � was studied on another occasion and the results are reported elsewhere %��&�Other parameters in the tests were the presence or lack of railings on the top surfaceof the deck and the angle of wind incidence �� to '��

Figure �� The centre portion of the section models �dimensions in mm��

Figure �� The various widthtodepth ratios investigated�

The pressure taps were positioned in an equidistant manner chordwise so thatthe contributory area of each of the taps was equivalent� For the larger widthtodepth ratios� up to �� pressure taps were �tted per strip� Twotoone pneumaticmanifolding of the surface pressures was thus performed for the models with B�D�� and �� A veri�cation of the eect of the local pressure averaging was madeby measuring the surface pressures for each of the �� taps of a strip of the B�D�� model and comparing the results with twotoone manifolded pressures� Mean�rms and power spectral densities of the integrated vertical and torsional forces werefound to match within a �( error margin�

The �� pressure transducers were sampled at a frequency of �� Hz for � periodsof �� seconds� Data reduction consisted of digital lowpass �ltering� integration


mean pressure distribution

rms pressure distribution

95146jm0.032

Flow

Figure �� Vector �eld representation of surface pressure distribution on B�D� ��medium spire exposure� with railings� �� ms�� Scaling� depth of deck� mean Cp of� and rms Cp of �� Polarity� negative away from the surface� The rms distributionis given for the projected tributary area�

of surface pressures to form timevarying force coe�cients �vertical� torsional andalongwind� and spectral analysis� Crosssectional admittances and spanwise coherence of wind forces were calculated for each case� Figs� � and � show examplesof surface pressure distributions in turbulent ow�



95146jm0.239

Flow

Figure �� Vector �eld representation of surface pressure distribution on B�D� ��large spire exposure� with railings� �� ms�� Scaling� depth of deck� mean Cp of �and rms Cp of �� Polarity� negative away from the surface�


�� Flow investigations

Three turbulent ow conditions were investigated� two generated by spires �� mupstream of the model and one by a coarse grid placed �� m upstream� The twospire sets gave similar length scales of turbulence but dierent turbulence intensitieswhile the position of the grid was selected to match the turbulence intensity of oneof the spire sets but giving much smaller length scales�

The grid had an isotropic mesh of �� m with a bar size of �� m� The mediumspires� in a set of three placed �� m upstream of the model� where �� m wide atthe base� �� m at the top and �� m high� while the large spires where �� m atthe base� �� m at the top and �� m high� The tests in the wind tunnel weretypically conducted at a mean wind speed of �� ms��

The turbulent ow characteristics were de�ned using hotwire anemometry �Dantec�s StreamLine and StreamWare system�� Typically� two sets of bidimensionalprobes �Dantec ��P�� were used to measure the u and w components of the wind uctuations� and subsequently were rotated �� to measure the v component� Thetests were carried out at the section model position� without the model in place� Atotal of � spanwise separations of the probes were investigated to de�ne the ow�eld in terms of onepoint and twopoint statistics� The separations ranged from�� mm to �� mm �i�e� �� mm� �� mm� �� mm� �� mm� �� mm� �� mm� ��mm� �� mm�� The sampling frequency was set at �� Hz for a sampling time of�� seconds� A �thorder Butterworth low pass �lter set at �� Hz was applied tothe signal before data acquisition�

� Analysis of the �ow conditions

�� Isotropic turbulence model and the von K�arm�an spectrum

In wind tunnels� the turbulence behind grids or large spires has an isotropic character %�&� This is especially true for locations away from the wind tunnel boundaries�It is thus appropriate to compare the ow conditions of this study with an isotropicturbulence model based on the von K�arm�an spectrum� A detailed discussion of theapplicability of isotropic turbulence model in wind engineerring can be found in%�� &�

Here� the isotropic tensor given by Mann in %��&� based on the von K�arm�anenergy spectrum� was �tted to the measured wind spectra for the u� v and w windcomponents� The onepoint spectrum for the u component can be expressed by�

Su�k��

��

��

��L�� ' k��

� ��

��

and for the v and w components�


0.1 1 3 10 30 10010−3

10−2

10−1

k1

S(k

1)

(a)

variance= 0.635

0.1 1 3 10 30 100

10−2

10−1

k1

k1 S

(k1)

(b)

Figure �� Graphical representation of Lxu and L for the von K�arm�an spectrum of u�

�� with L� �� and �� a� Lx

u � �Su�� m$ b� L is the inverse of

k� corresponding to the maximum of k�Su�k�� L� �� m�

Sv�w�k��

��

��L�� ' �k��L�� ' k��

� ��

��

Here� k� is a wave number ��f�V �� represents the rate of viscous dissipationof turbulent kinetic energy� � is the von K�arm�an constant and L is a length scalecommon to the three components of the spectral tensor�

Relationships exist between L and the integral length scales Lxu� L

xv and Lx

w forisotropic turbulence %��&�

Lxu � L

�p�

#��

#��

�L ��

��

�uSu�k� � ��

Lxv�w �

�

�Lxu �

�

�L ��

Equation �� expresses that the integral length scales �also de�ned as Lxu �R

�

� Ruu�x�dx for example� are related to the wind spectrum at frequency zero�The physical meaning of Lx

u is not obvious and it is considered here� as in %��&� thatthe wave length� denoted L� associated with the peak of k�Su�k�� or fSu�f� is amore appropriate characteristic length to describe a turbulent ow �eld�L represents the length of the eddies that carry most of the energy of the tur

bulence and it will be used as a reference length later in this study� Fig� � de�nesgraphically Lx

u and L�


L is related to L for the von K�arm�an spectrum by�

Lu �

��

�

� ��

L � ��L ��

Lv�w ��

�� ' �p��

��

L � ��L ��

The variances for isotropic turbulence are all equal and can be expressed %��& by�

�u�v�w ��

��

#��

�#��

�p�� L ��

��L

��

�� Two�point statistics� the span�wise coherence

Analytical derivations of crossspectra based on the von K�arm�an spectrum existin various forms� notably due to Irwin %�& and Thompson %�&� This constitutes aclear advantage in using the isotropic turbulence model� Here the more completederivations due to Mann et al� %��& is used to de�ne the spanwise coherence of theoncoming ow uctuations�

The coherence function between points � and � is expressed by the magnitudesquared of the real and imaginary parts of the crossspectrum divided by the productof the onepoint spectra at � and �� For isotropic turbulence� the imaginary partof the crossspectrum� Qu� is zero� so that only the real part� Co� is necessary todescribe the coherence as a function of k� and spanwise separation "y�

cohu�"y� k�� Co��"y� k��

S�u�k��

��

coh��u �"y� k�� Co�"y� k��

Su�k��

Equation �� is also referred to as the normalized cospectrum %�& or cocoherence�cocoh� For any separation and length scale L� the cocoherences are expressed in%��& by�

cocohu�"y� k��

#��

� ��

�

� ��K��

�

��K��

��

cocohv�"y� k��

#��

� ��

�

� ��

K�� '

��"yk��

�� ' ��"yk��K��

��

cocohw�"y� k��

#��

� ��

�

� ��

��K��

��yL

�� ' ��"yk��

�K��

��


0.1 0.5 1 3 5 10 50

10−4

10−3

k1

k1 S

_u,v

,w(k

1) /V

^2

u

0.1 0.5 1 3 5 10 50

10−4

10−3

k1

v

0.1 0.5 1 3 5 10 50

10−4

10−3

k1

w

Figure �� Averaged autospectra of the wind uctuations normalized by the meanwind velocity squared for the exposures of this study� Solid lines� large spires$dashed lines� medium spires$ dotted lines� grid�

where K�� are modi�ed Bessel functions of the second kind and�

� �

�"y�k�� '

"y�

L�

� ��

� k�"y

s� '

�

k��L�

��

is the von K�arm�an collapsing parameter�

�� Least square �t of wind data

Onepoint spectra of the wind uctuations in u� v and w were calculated for eachof the three exposures� For each case� a total of �� spectra measured at dierentspanwise loactions at the section model position were averaged to de�ne a universalspectra for each exposure� Fig� � presents the averaged onepoint spectra� The gridexposure has its maximum at wave numbers almost � times larger than the spireexposures� implying � times smaller scales of turbulence�

To characterise the ow conditions at the section model position� the averagedspectra were �tted with the von K�arm�an model� �� and �� leaving L and ��

�� as

oating parameters�The least square �t was made with a Simplex search method %�&� part of the

Matlab analysis software� Figs� � compares the experimental data to the �tted vonK�arm�an spectra� while Table � presents the results of the �ts and summarises thecharacteristics of the turbulence for the three exposures�

The turbulence intensities Iu�v�w were de�ned as the standard deviation of thevelocity uctuations of a given component divided by the mean of u� that is �V �


0.1 0.5 1 3 5 10 50

10−2

10−1

k1

k1 S

(k1)

u

L= 0.41

0.1 0.5 1 3 5 10 50

10−2

10−1

k1

v

L= 0.46

0.1 0.5 1 3 5 10 50

10−2

10−1

k1

w

L= 0.39

Figure �� Least square �t of the von K�arm�an model� �� and �� for the averaged

spectra for the medium spire exposure� The �tted parameters were L and ��

Table �� Characteristics of ow conditions for the three exposures based on theleast square �t of the von K�arm�an spectrum�

Expo� u�component v�component w�component

sures Iu L ��

� Lu Iv L ��

� Lv Iw L ��

� Lw

�� m� �m��

s� m� �� m� �m

��

s� m� �� m� �m

��

s� m�

Large �� spiresMedium �� spiresGrid ��

The standard deviations were corrected to include the contribution of the higher

frequencies �� (� by extrapolating the wind spectra using a k�� turbulence

decay relationship�If the ow conditions were truly isotropic� the turbulence intensities as well as

the length scales L would have been the same for the three wind components� Thiswas not observed here for the spire exposures where the v component appeared tohave larger characteristics� However� the grid exposure had quasiisotropic characteristics�

Spanwise cocoherence� A similar least square �t analysis was performed forthe cocoherence data� using the equations describing the twopoint statistics forthe istropic turbulence model presented in the earlier section� The cocoherencedata were calculated by fast Fourier transforms �FFT� of the time series of the


wind uctuations� sampled at �� Hz for �� seconds� Averaging the �� blocksof ��point FFT resulted in cocoherence spectral estimates with a coe�cient ofvariation of ��(�

Each of the cocoherence sets of data for eight spanwise separations� from �� mmto �� mm� were �tted with equations �� to �� The median of the resultingeight von K�arm�an length scales L� one per separation� was calculated and selectedas an adequate representation of Lcoh

� for a given wind component and exposure�For smaller separations� which is approaching isotropic turbulence conditions� thedispersion in the results of the �ts of Lcoh was very limited compared to the dispersion obtained for the larger separation� The median Lcoh for each case is givenin Table � and a comparison between analytical calculations of cocoherence andexperiments are given on Figs� �� and �� for � separations� respectively for themedium spire exposure and the grid exposure�

The isotropic turbulence model �tted well the spanwise cocoherence measurements in most of the cases� even though the characteristics of the onepoint spectrumshowed that the ow conditions were not entirely isotropic� The results of the �tof the w component showed lower values than for u and v� It appeared that theanisotropy of the ow aected more the w component�

Table �� Least square �t of the von K�arm�an length scale Lcoh for the cocoherencedata for the three exposures�

Exposures ucomponent vcomponent wcomponentLcoh Lcoh Lcoh

�m� �m� �m�

Large spires �� Medium spires �� Grid ��

Collapsing of the cocoherence data� To verify the use of � as a collapsingparameter� the spanwise cocoherence data for the three exposures for all separations were plotted versus � �� Fig� �� illustrates clearly that all data for themedium spires collapsed well on one line for u� v and w�

An empirical expression� orginally used in %�&� was �tted to the collapsed data�This expression �� is similar to a Weibull function and will be used later on forfurther analysis of the spanwise cocoherence� The results of the �t are given inTable ��

cocoh�� exp %�c��c� & cos�c��

�The subscript coh was introduced to di�erentiate the length scale obtained from a t of theco�coherence data� Lcoh� with the length scale L obtained from a t of the one�point spectrum�For isotropic conditions Lcoh � L�


0 20 40 60 80

0

0.5

1delta y= 0.03 m

Co−

coh,

U

L= 0.44

0 20 40 60 80

0

0.5

1

Co−

coh,

V

L= 0.4

0 20 40 60 80

0

0.5

1

Co−

coh,

W

f (Hz)

L= 0.27

0 20 40 60 80

0

0.5

1delta y= 0.06 m

L= 0.44

0 20 40 60 80

0

0.5

1L= 0.4

0 20 40 60 80

0

0.5

1

f (Hz)

L= 0.27

0 20 40 60 80

0

0.5

1delta y= 0.09 m

L= 0.44

0 20 40 60 80

0

0.5

1L= 0.4

0 20 40 60 80

0

0.5

1

f (Hz)

L= 0.27

0 20 40 60 80

0

0.5

1delta y= 0.24 m

L= 0.44

0 20 40 60 80

0

0.5

1L= 0.4

0 20 40 60 80

0

0.5

1

f (Hz)

L= 0.27

Figure �� Variations of cocoherence as a function of frequency� separations andwind components for the medium spire exposure� The solid lines on the graphs arethe results of the least square �t of the experimental data with �� to �� withLcoh as a oating parameter�


0 20 40 60 80

0

0.5

1delta y= 0.03 m

Co−

coh,

U

L= 0.2

0 20 40 60 80

0

0.5

1

Co−

coh,

V

L= 0.18

0 20 40 60 80

0

0.5

1

Co−

coh,

W

f (Hz)

L= 0.12

0 20 40 60 80

0

0.5

1delta y= 0.06 m

L= 0.2

0 20 40 60 80

0

0.5

1L= 0.18

0 20 40 60 80

0

0.5

1

f (Hz)

L= 0.12

0 20 40 60 80

0

0.5

1delta y= 0.09 m

L= 0.2

0 20 40 60 80

0

0.5

1L= 0.18

0 20 40 60 80

0

0.5

1

f (Hz)

L= 0.12

0 20 40 60 80

0

0.5

1delta y= 0.24 m

L= 0.2

0 20 40 60 80

0

0.5

1L= 0.18

0 20 40 60 80

0

0.5

1

f (Hz)

L= 0.12

Figure �� Variations of cocoherence as a function of frequency� separations andwind components for the grid exposure� The solid lines on the graphs are theresults of the least square �t of the experimental data with �� to �� with Lcoh

as a oating parameter�


0 2 4 6 8 100

0.5

1C

o−co

h, U

Collapsing parameter, with L= 0.44

c1,c2,c3: 0.83, 1.16, 0.61

0 2 4 6 8 100

0.5

1

Co−

coh,

V


c1,c2,c3: 0.4, 1.32, 0

0 2 4 6 8 100

0.5

1

Co−

coh,

W


c1,c2,c3: 0.73, 1.03, 0.27

Figure �� Variations of cocoherence as a function of the collapsing parameter ��for the medium spire exposure�

Table �� Least square �t of the collapsed cocoherence data for the three exposuresfor all separations� c�� c� and c� are the �tted parameters�

Expo ucomponent vcomponent wcomponentsures c� c� c� c� c� c� c� c� c�Large �� spiresMedium �� spiresGrid ��


�� The Fichtl�McVehil spectrum

It was observed in the earlier section that the onepoint spectrum of the isotropicturbulence model did not �t adequately the experimental data for the three exposures� The shape of the measured spectra seemed less peaky� indicating some shearin the ow %��&� and the �t seemed to be pulled towards the higher wave numbers�

To improve the �t of the spectra from the wind tunnel wind data� the FichtlMcVehil spectrum %��& was chosen for its exibility� This spectral shape has theadvantage of having a oating parameter r that can be adjusted to match the peakiness of the spectrum� while the other oating parameter fm is adjusted to matchthe position of the peak on the abscissa� In fact fm corresponds to the frequencyin Hz at which fSu�f� has a maximum� It is thus very helpful to determine L�

The FichtlMcVehil spectrum is of the form�

fSu�v�w�f�

�u�v�w�

a�

ffm

�� ' ��

�ffm

�r� ��r

��

where

a ��

�r r #

��r

�#�

��r

�#��r

� � ��

Note that when r � �� is equivalent to the von K�arm�an spectrum of the ucomponent and when r � �� has the Kaimal spectral shape %�&�

The onepoint spectra described in the previous section were �tted in a leastsquares manner with �� keeping r and fm as oating parameters� Examples ofthe �t for the medium spires and the grid exposures are given in Figs� �� and ��Table � gives a summary of the results�

The length scales� L� corresponding to the peak of the spectrum appeared tobe slightly higher using the FichtlMcVehil spectral shape compared to the vonK�arm�an model�

Table �� Least square �t of the FichtlMcVehil spectrum for the three exposures�

Expo ucomponent vcomponent wcomponentsures r fm Lu r fm Lv r fm Lw

�Hz� �m� �Hz� �m� �Hz� �m�

Large spires �� Medium spires �� Grid ��


0.1 0.5 1 3 5 10 50

10−2

10−1

k1

k1 S

(k1)

u

r = 1.12

fm= 3.63

0.1 0.5 1 3 5 10 50

10−2

10−1

k1

v

r = 1.24

fm= 5.17

0.1 0.5 1 3 5 10 50

10−2

10−1

k1

w

r = 1.44

fm= 6.54

Figure �� Least square �t of the FichtlMcVehil spectrum �� for the averagedspectra for the medium spire exposure� The �tted parameters were r and fm�

0.1 0.5 1 3 5 10 50

10−2

10−1

k1

k1 S

(k1)

u

r = 1.04

fm= 9.16

0.1 0.5 1 3 5 10 50

10−2

10−1

k1

v

r = 1.35

fm= 15.56

0.1 0.5 1 3 5 10 50

10−2

10−1

k1

w

r = 1.55

fm= 18.19

Figure �� Least square �t of the FichtlMcVehil spectrum �� for the averagedspectra for the grid exposure�


� Characteristics of the unsteady wind forces

�� Cross�sectional forces and aerodynamic admittance

The aerodynamic lift forces induced by gusty winds on a crosssectional strip of abridge deck can be described �see Part IA� by�

Fz�b�t� � � �V B

�

�Czu�t� ' C �

zw�t�

��

The bueting lift forces are proportional to w and intuitively it can be deducedthat the duration of only the larger gusts will be su�ciently long to generate variations of surface pressures and thus lift variations� Oncoming ow with larger lengthscales will then be more e�cient in generating unsteady lift on a strip than owwith the same turbulence intensity but smaller length scales� This is illustratedin Fig� �� where the spectral densities of the lift and torque coe�cients� Cz�m� arecompared for two exposures keeping all the other test parameters constant� Thespectra showed a much larger magnitude for the medium spire exposure than forthe grid exposure�

0.1 0.5 1 3 5 10 50

10−4

10−3

10−2

k1

k1 S

(k1)

lift

0.1 0.5 1 3 5 10 50

10−5

10−4

10−3

k1

k1 S

(k1)

torsion

Figure �� Power spectral density of Cz�m versus k� for the medium spire exposure�solid line� and the grid exposure �dotted line�$ B�D � �� V �� ms�� and Iw�� , �� (�

This concept is expressed in the frequency domain by the aerodynamic admittance and was introduced for bridge decks by Davenport �see Part I�� The ratiobetween the scales of turbulence and the deck width� L�B� is thus an importantparameter to characterise the aerodynamic admittance� Table � presents L�B forthe test conditions of this study�


Table �� Ratio of the turbulence length scales to the deck width� L�B for the threeexposures and the three aspect ratios�

Large spires Medium spires Gridu v w u v w u v w

B�D�� B�D�� B�D��

The aerodynamic admittance of the various deck con�gurations was calculatedby dividing the spectrum of the force coe�cients� SCz�m � on one chordwise strip bythe spectrum of the oncoming wind velocity uctuations� Su�w� For the case of thelift forces� the admittance can be expressed by�

jAz �f� j� ��V � SCz �f�

�C�z Su�f� ' C �

z� Sw �f�

��

where C �z is the derivative near �� of the lift coe�cient Cz with respect to the angle

of wind incidence in radians� Table � illustrates typical variations of the staticcoe�cients and their derivatives with increasing widthtodepth ratio�

Table �� Time averaged aerodynamic force coe�cients for the medium spire exposure� deck without railings �based on B��

Cz�� Cm�� C �

z�� C �

m��

B�D�� B�D�� B�D��

Fig� �� presents results of the aerodynamic admittance measurements for fourcases� two aspect ratios and two wind conditions� The vertical turbulence intensitywas the same for each case� but the ratio Lw�B was varied from �� to �� Itindicates that the larger the ratio between the size of the gusts and the deck width�the larger the admittance�

Fig� �� b� compares the measured admittance for B�D� � and the mediumspire exposure to the Liepmann�s approximation to the Sears function analyticallyderived for a thin airfoil with a lift slope of �� %�&� The ordinate is the product of C �

z�

and j Az�f� j�� The measured admittance never produced more lift than the lineartheory predicted for a thin airfoil and showed much less admittance at low reducedfrequency� It can also be said that even though the deck crosssection studied was


a poorer lift generator� it could generate as much lift than a thin airfoil at higherreduced frequency� Bodyinduced turbulence might explain this behaviour�

For this case� the test conditions were representative of full scale conditions wherea ratio Lw�B � �� m )�� m is frequently seen� A thorough analysis of the aerodynamic admittance measurements is presented in Part IIB�

0.001 0.01 0.1 10.01

0.1

1

fB/V

Aer

odyn

amic

adm

ittan

ce

(a)

B/D= 5, L_w/B= 1.50

B/D= 10, L_w/B= 0.80

B/D= 12.67, L_w/B= 0.58

B/D= 12.67, L_w/B= 0.23

0.001 0.01 0.1 11

10

fB/V

Aer

odyn

. Adm

it. x

Cz‘

*2

Liepmann approx.

(b)

Figure �� a� Variations of lift aerodynamic admittance with reduced frequency forfour cases in turbulent ow� b� Comparisons of the product of the admittance andlift slope squared� for B�D� � with railings� medium spire� Lw�B� �� dashedline�� with theoretical predictions for a thin airfoil �solid line� %�&�

�� Span�wise cross�correlation and co�coherence

The spanwise variations of the unsteady wind forces acting on the deck can beexpressed by the cocoherence function �as de�ned in Section �� between forcesmeasured simultaneously on two distinct chordwise strips� A total of six spanwiseseparations were analysed for each case� varying from �� mm to �� mm� Similarmeasurements to characterize the oncoming ow �eld were reported in the previoussection �separations from �� mm to �� mm��

Figs� �� and �� illustrate the variations of the cocoherence of the forces with frequency for four spanwise separations and various test conditions� It was observedthat for a given wind exposure� the larger the deck width� the larger the spanwisecoherence$ and that for a given deck width� the smaller the scales of turbulencethe smaller the cocoherence� While the latter observation was not surprising� theformer indicated that the application of the strip assumption could have some limitations for the bridge decks studied�


0 10 20 30 40 50 60 70 80

0

0.2

0.4

0.6

0.8

1

delta_y.: 0.06 mCo−

cohe

renc

e

0 10 20 30 40 50 60 70 80

0

0.2

0.4

0.6

0.8

1

delta_y.: 0.13 m

0 10 20 30 40 50 60 70 80

0

0.2

0.4

0.6

0.8

1


cohe

renc

e

f (Hz)0 10 20 30 40 50 60 70 80

0

0.2

0.4

0.6

0.8

1

delta_y.: 0.3 m

f (Hz)

Figure �� Variations of the cocoherence of the lift forces with frequency� spanwiseseparations and bridge deck aspect ratios for the medium spire exposure� Dottedline� B�D � �$ solid line� B�D� �� and dashed line� B�D� ��

0 10 20 30 40 50 60 70 80

0

0.2

0.4

0.6

0.8

1


cohe

renc

e

0 10 20 30 40 50 60 70 80

0

0.2

0.4

0.6

0.8

1

delta_y.: 0.13 m

0 10 20 30 40 50 60 70 80

0

0.2

0.4

0.6

0.8

1


cohe

renc

e

f (Hz)0 10 20 30 40 50 60 70 80

0

0.2

0.4

0.6

0.8

1

delta_y.: 0.3 m

f (Hz)

Figure �� Variations of the cocoherence of the lift forces with frequency� spanwiseseparations and wind exposures for B�D � �� Dotted line� grid exposure$ solidline� medium spire exposure and dashed line� large spire exposure�


�� Crosscorrelation coe�cient

The purpose of the analysis that follows is to compare the twopoint statistics of theinput �the wind uctuations� to the characteristics of the output �the aerodynamicforces�� The �rst parameter to compare is the spanwise crosscorrelation coe�cientde�ned as�

R��"y� � ��

��

where �� is the covariance between points � and � separated by "y� R�� is anindicator of the correlation of the process for the entire frequency range �zero timelag� and it is thus only a function of the separation�

The crosscorrelation coe�cients were calculated for the various tests conditions and a summary of the results is given in Figs� �� to �� For all cases� thecrosscorrelation of v was the largest of the three wind components� and the crosscorrelation of the forces was always considerably larger than the oncoming wind�Fig� �� Also� it was observed that the larger the deck width� the larger thecrosscorrelation� for all exposures�

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

delta y (m)

Cor

rela

tion

coef

ficie

nt

Wind (medium spires) and vertical forces

w

u

v

forces, B/D=12.67

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

delta y (m)

Cor

rela

tion

coef

ficie

nt

Wind (grid exposure) and vertical forces

w

uv

forces, B/D=12.67

Figure �� Variations of the spanwise crosscorrelation coe�cient of the verticalaerodynamic forces �solid line� for B�D�� and the wind velocity uctuations�dashed lines� for the medium spire exposure and the grid exposure�

Figs� �� and �� compare the correlation of the forces for dierent B�D ratios asa function of the normalized separation "y�B� The curves tend to collapse on oneline for the lower values of "y�B but for the higher values� the curve associatedwith the smaller deck width showed larger correlation� That is to say that themodel with the smaller B�D ratio showed a larger correlation length in relation toits width than the other models� Since the narrower model has the larger Lcoh�Bratio� this observation is consistent with earlier �ndings %�& that RFF increases withan increase of Lcoh�B �see Part IB��


The larger crosscorrelation of the forces than the velocity uctuations was to beexpected since the process to generate the forces implies that only the larger scalegusts will create lift� The structure does some conditional sampling� selecting onlythe more correlated part of the wind� However� this can only explain a small partof the dierence between Rww and RFF � It also suggests the nonvalidity of thestrip assumption�

0 0.5 1 1.5 20.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

medium spires exposure

delta_y / B

Cor

rela

tion

coef

ficie

nt

Vertical forces

B/D=5 B/D=10 B/D=12.67

0 0.5 1 1.5 20.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

delta_y / B

Cor

rela

tion

coef

ficie

nt

Torsional forces

Figure �� Variations of the spanwise crosscorrelation coe�cient of the aerodynamic forces as a function of normalized separations "y�B for the medium spireexposure�

0 0.5 1 1.5 20.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

grid exposure

delta_y / B

Cor

rela

tion

coef

ficie

nt

Vertical forces

B/D=5 B/D=10 B/D=12.67

0 0.5 1 1.5 20.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

delta_y / B

Cor

rela

tion

coef

ficie

nt

Torsional forces

Figure �� Variations of the spanwise crosscorrelation coe�cient of the aerodynamic forces for the grid exposure�


�� Cocoherence� wind velocity versus forces

A comparison of the cocoherence of the lift forces for two aspect ratios to the cocoherence of the velocity uctuations in v and w is given in Fig� �� The comparisonsare made for two separations� The cocoherence of the forces is clearly larger thanthe cocoherence of w but is similar to the cocoherence of v� This similarity ispossibly only coincidental�

0 20 40 60 80

0

0.5

1

Co−

cohe

renc

e

Wind, w−component

sep.: 60 mm

0 20 40 60 80

0

0.5

1

Frequency (Hz)

Co−

cohe

renc

e sep.: 240 mm

0 20 40 60 80

0

0.5

1Wind, v−component

sep.: 60 mm

0 20 40 60 80

0

0.5

1

Frequency (Hz)

sep.: 240 mm

0 20 40 60 80

0

0.5

1Lift Forces, B/D= 5:1

sep.: 60 mm

0 20 40 60 80

0

0.5

1

Frequency (Hz)

sep.: 240 mm

0 20 40 60 80

0

0.5

1Lift Forces, B/D= 10:1

sep.: 60 mm

0 20 40 60 80

0

0.5

1

Frequency (Hz)

sep.: 240 mm

Figure �� Variations of the spanwise cocoherence of the forces and of the velocity uctuations for two separations as a function of frequency�

To verify this last aspect and to describe in details the relationships between thecocoherence of the wind and the resulting forces� curves showing the variations ofthese quantitites with the separation to the wave length ratio� k�"y� are given inFigs� �� to �� All the test cases are represented on these �gures for the con�gurationwithout railings and a mean wind speed of �� ms�� The cocoherence of the wind uctuations is represented by the von K�arm�an model� equations �� and �� withLcoh obtained by a least square �t of the cocoherence data �see Section ��

Fig� �� shows results for the case where the Lcoh�B was the largest for this studyand is representative of fullscale conditions� For spanwise separations larger than��( of Lw� the cocoherence of the forces is much larger than the cocoherence of wand equivalent to the cocoherence of v for k�"y smaller than �� The strip assumption is thus clearly not valid for this case �and for all the subsequent cases�� However�for the smaller spanwise separation where "y�Lw� ��(� the cocoherence of w andthe lift forces were found to be equivalent for the abovementioned test case�


Fig� �� presents the results for another test case representative of fullscale conditions� It is for the large spire exposure and the B�D � �� Here also� thecocoherence of lift is much larger than the cocoherence of w and somewhat largerthan the cocoherence of v� This �gure shows that the observation made earlierassociating cocoherence of v and lift force was coincidental except for small k�"y�

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.03 m

Co−

cohe

renc

e

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1delta_y.: 0.13 m

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1delta_y.: 0.24 m

k1 delta_y

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1sep.: 0.3 m

k1 delta_y

Figure �� Cocoherence as a function of k�"y for the lift forces �dotted line� forB�D � � and for the wind for the medium spire exposure� Solid line� v withLcoh� �� and L�� m$ dashed line� w with Lcoh� �� and L�� m�


0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.03 m

Co−

cohe

renc

e

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1delta_y.: 0.13 m

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1delta_y.: 0.24 m

k1 delta_y

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1sep.: 0.3 m

k1 delta_y

Figure �� Cocoherence as a function of k�"y for the lift forces �dotted line� forB�D � �� and for the wind for the medium spire exposure� Solid line� v withLcoh� �� and L�� m$ dashed line� w with Lcoh� �� and L�� m�

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.03 m

Co−

cohe

renc

e

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1delta_y.: 0.13 m

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1delta_y.: 0.24 m

k1 delta_y

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1sep.: 0.3 m

k1 delta_y

Figure �� Cocoherence as a function of k�"y for the lift forces �dotted line�� B�D� �� and for the wind for the medium spire exposure� Solid line� v withLcoh� �� and L�� m$ dashed line� w with Lcoh� �� and L�� m�


0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.03 m

Co−

cohe

renc

e

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1delta_y.: 0.13 m

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1delta_y.: 0.24 m

k1 delta_y

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1sep.: 0.3 m

k1 delta_y

Figure �� Cocoherence as a function of k�"y for the lift forces �dotted line� forB�D � � and for the wind for the grid exposure� Solid line� v with Lcoh� ��and L�� m$ dashed line� w with Lcoh� �� and L�� m�

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.03 m

Co−

cohe

renc

e

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1delta_y.: 0.13 m

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1delta_y.: 0.24 m

k1 delta_y

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1sep.: 0.3 m

k1 delta_y

Figure �� Cocoherence as a function of k�"y for the lift forces �dotted line� forB�D �� and for the wind for the grid exposure� Solid line� v with Lcoh� ��and L�� m$ dashed line� w with Lcoh� �� and L�� m�


0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.03 m

Co−

cohe

renc

e

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1delta_y.: 0.13 m

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1delta_y.: 0.24 m

k1 delta_y

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1sep.: 0.3 m

k1 delta_y

Figure �� Cocoherence as a function of k�"y for the lift forces �dotted line� forB�D � �� and for the wind for the grid exposure� Solid line� v with Lcoh�� and L�� m$ dashed line� w with Lcoh� �� and L�� m�

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.03 m

Co−

cohe

renc

e

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1delta_y.: 0.13 m

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1delta_y.: 0.24 m

k1 delta_y

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1sep.: 0.3 m

k1 delta_y

Figure �� Cocoherence as a function of k�"y for the lift forces �dotted line� forB�D � �� and for the wind for the large spires exposure� Solid line� v withLcoh� �� and L�� m$ dashed line� w with Lcoh� �� and L�� m�


0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.03 m

Co−

cohe

renc

e

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1delta_y.: 0.13 m

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1delta_y.: 0.24 m

k1 delta_y

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1sep.: 0.3 m

k1 delta_y

Figure �� Cocoherence as a function of k�"y for the lift forces �dotted line� forB�D � �� and for the wind for the large spires exposure� Solid line� v withLcoh� �� and L�� m$ dashed line� w with Lcoh� �� and L�� m�

�� In uence of the Lw�B ratio

To characterize the in uence of the Lw�B ratio on the cocoherence of the forces�a series of comparisons are presented in Figs� �� to �� In theory� test conditionswith similar Lw�B ratios should show similar cocoherence curves� This is veri�edqualitatively in Fig� �� For all separations� the cocoherence of the lift for theB�D�� and the medium spire exposure is equivalent to the B�D�� case withthe large spire exposure�

For a more rigorous analysis� in addition to the geometric similarity representedby Lw�B� the comparisons should be based on the normalized separation "y�B andthe wavelength to the deckwidth ratio k�B� For similar Lw�B the comparisonsbetween dierent deck widths B� and B� should verify the following expression�

cocohFF �k�B��"y�B�� cocohFF �k�B��"y�B��

Fig� �� presents a comparison for the same test conditions as Fig� �� but usingthe extended geometric similarity described above� The agreement between thesetwo cases is excellent�

When the test conditions of Fig� �� are reversed� one would expect that the combination of the larger deck width and the medium spires will have smaller cocoherencethan the combination of smaller deck width with the larger spires� Surprisingly� this


0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.03 m

Co

−co

he

ren

ce

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1delta_y.: 0.13 m

Co

−co

he

ren

ce

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1delta_y.: 0.24 m

k1 delta_y

Co

−co

he

ren

ce

0 1 2 3 4

0

0.5

1sep.: 0.3 m

k1 delta_y

Figure �� Comparisons of the cocoherence of the lift forces for two test conditions�B�D � �� dotted line� with the medium spire exposure� Lw�B �� and B�D� �� solid line� with the large spire exposure� Lw�B ��

was veri�ed only for very small k�B �smaller than �� as presented in Fig� �� Forlarger k�B� the two curves collapsed indicating that the test case with the largerdeck width showed more cocoherence than anticipated�

Fig� �� reports two cases where Lw�B � �� The case with the larger deckwidth showed considerably larger cocoherence than the case with the narrowerdeck in grid turbulence� This con�rms the trend of the latter case and suggeststhat the spanwise cocoherence of the forces is in uenced not only by the Lw�Bratio but also directly by the deck width itself� Note that for this case the exposureshave similar turbulence intensity but the grid turbulence has a much larger contentof small scale turbulence that might play a role in the process�

A comparison where one of the cases would clearly not satisfy the condition ofthe strip assumption� stating that the scale of turbulence should be bigger thanthe characterisitic length of the body is presented in Fig� �� It is for B�D� ��and the grid exposure so that Lw�B � �� which is compared to a Lw�B � ��case� As expected� the Lw�B � �� case showed cocoherence curves larger thanthe Lw�B � �� case up to k�B values of �� For larger k�B the Lw�B � ��case showed larger cocoherence� It is likely that the secondary spanwise ows areimportant for the Lw�B � �� case� increasing the cocoherence drastically� It is


0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

delta_y/ B: 0.4333

Co−

cohe

renc

e

k1 B0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

delta_y/ B: 0.8

k1 B

Figure �� Same test conditions as Fig� �� but the comparison is made as a functionof normalized separations "y�B and the wavelength to the deckwidth ratio k�B�B�D � �� and Lw�B �� dotted line$ B�D � �� and Lw�B �� solidline�

believed that very early� as the approaching vortices are passing the bridge deck�they are getting reorganized by the deck itself in a more coherent manner� eventhough the deck remains stationary�

Finally� Fig� �� compares two cases where Lw�B � �� The test conditionswith the larger deck width has larger cocoherence for all normalized separationsand all values of k�B�

0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

delta_y/ B: 0.4333

Co−

cohe

renc

e

k1 B0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

delta_y/ B: 0.8

k1 B

Figure �� Comparisons of the cocoherence of the lift forces for two test conditions�B�D � �� dotted line� with the large spire exposure� Lw�B �� and B�D �� solid line� with the medium spire exposure� Lw�B ��


0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

delta_y/ B: 0.4211

Co−

cohe

renc

e

k1 B0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

delta_y/ B: 0.7895

k1 B

Figure �� Comparisons of the cocoherence of the lift forces for two test conditions�B�D � �� dotted line� with the medium spire exposure� Lw�B �� and B�D� � �solid line� with the grid exposure� Lw�B ��

0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

delta_y/ B: 0.4

Co−

cohe

renc

e

k1 B0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

delta_y/ B: 0.8667

k1 B

Figure �� Comparisons of the cocoherence of the lift forces for two test conditions�B�D � � �dotted line� with the medium spire exposure� Lw�B �� and B�D �� solid line� with the grid exposure� Lw�B ��


0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

delta_y/ B: 0.2Co−

cohe

renc

e

0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

delta_y/ B: 0.4333

0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

delta_y/ B: 0.8Co−

cohe

renc

e

k1 B0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

delta_y/ B: 1

k1 B

Figure �� Comparisons of the cocoherence of the lift forces for two test conditions�B�D � �� dotted line� with the medium spire exposure� Lw�B � �� and B�D� � �solid line� with the grid exposure� Lw�B � ��

�� In uence of the railings

Experience has shown that conventional road equipment such as crash barriers�median dividers and wind breaks can change signi�cantly the aerodynamics of abridge deck� To study this aspect� the presence or lack of railings on the deck waskept as a test parameters throughout this study� All the results reported previouslyreferred to the without railing con�guration�

Railings made of thin brass rods� � ��( porous with an overall height equalto ��( of the deck depth� were mounted at the leading and trailing edges of thetop surface of the section models� The railings represented a crash barrier with arelatively large porosity commonly used on bridge decks in Europe�

Based on observations made during this study� the in uence of the railings on thedynamic characteristics of the bueting wind loads can be summarised as follows�

�� Generally� an increase of spanwise coherence of the forces was observed whenrailings were �tted to the section models�

�� The increase of cocoherence for lift and overturning moment was compensatedfor by a reduction of the crosssectional forces on a given strip �tted withrailings�

�� The larger the deck width the lesser the in uence of the railings�


0.1 0.5 1 3 5 10 5010−4

10−3

k1

k1 S

(k1)

lift

0.1 0.5 1 3 5 10 5010−5

10−4

k1

k1 S

(k1)

torsion

Figure �� Power spectral density of Cz�m versus k� for the medium spire exposureand B�D � �� with railings �dotted line� and without railings �solid line�

�� The smaller the scales of turbulence the larger the increase of cocoherencefor the with railing case� especially for smaller separations�

The observations of point � and � are illustrated by Fig� �� and Fig� �� for themedium spire exposure and B�D� �� For the larger aspect ratio� B�D� �� thein uence of the railings appeared negligible� Fig� �� illustrates points � and �� Forsmall separations and smaller length scales� the spanwise coherence was found tobe larger with the railings in place�

By inspection of the unsteady pressure distribution �see Figs� �� and �� it wasobserved that the separation bubble was longer when the railings were �tted to thesection model� However� the large suction peaks observed on the top surface of thedeck near the leading edge were reduced by the presence of the railings� This couldexplain the observed reduction of the crosssectional forces on a strip �Fig� ��The elongation of the separation bubble due to the railings likely facilitates thegeneration of secondary cross ows that would increase the spanwise cocoherenceand crosscorrelation� This process is less important for the larger deck widths sincesecondary cross ows are probably already active and the spanwise cocoherenceis already large�


0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.03 m

Co

−co

he

ren

ce

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1delta_y.: 0.13 m

Co

−co

he

ren

ce

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1delta_y.: 0.24 m

k1 delta_y

Co

−co

he

ren

ce

0 1 2 3 4

0

0.5

1sep.: 0.3 m

k1 delta_y

Figure �� Cocoherence of the lift forces for the with railings �dotted line� andwithout railings �solid line� con�gurations� B�D � �� medium spire exposure�

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.03 m

Co

−co

he

ren

ce

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1delta_y.: 0.13 m

Co

−co

he

ren

ce

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1delta_y.: 0.24 m

k1 delta_y

Co

−co

he

ren

ce

0 1 2 3 4

0

0.5

1sep.: 0.3 m

k1 delta_y

Figure �� Cocoherence of the lift forces for the with railings �dotted line� andwithout railings �solid line� con�gurations� B�D � �� grid exposure�




95146jm0.205

Flow

Figure �� Surface pressure distribution on B�D� �� smooth ow� without railings�� ms�� Scaling� depth of deck� mean Cp of � and rms Cp of �� Polarity� negativeaway from the surface� The rms distribution is given for the projected tributaryarea�



95146jm0.214

Flow

Figure �� Surface pressure distribution on B�D� �� smooth ow� with railings� �ms�� Scaling� depth of deck� mean Cp of � and rms Cp of �� Polarity� negativeaway from the surface�


�� In uence of the angle of wind incidence

The angle of wind incidence was also chosen as a test parameter� The tests wereperformed by systematically changing the angle of attack of the motionless sectionmodels for the medium spire exposure� The range of angle covered was limitedbetween �� and '�� positive nose up��

In general� the positive angles of wind incidence gave rise to larger spanwise cocoherence� the separation bubble behing elongated on the top surface� As observedearlier the larger cocoherence was compensated for by smaller aerodynamic forceson a crosssectional strip� However� in opposition to the observations made duringthe study of the in uence of the railings� a larger in uence was found for the largerdeck widths� In fact� for the B�D�� case� no real dierence were observed in thecocoherence curves for the angles tested�

Fig� �� to Fig� �� summarise the in uence of the angle of wind incidence�

0.1 0.5 1 3 5 10 5010−4

10−3

k1

k1 S

(k1)

lift

0.1 0.5 1 3 5 10 5010−5

10−4

k1

k1 S

(k1)

torsion

Figure �� Power spectral density of Cz�m versus k� for the medium spire exposureand B�D � �� for three angles of wind incidence� Solid line� ��$ dotted line�'��$ dashed line� ��


0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

delta_y.: 0.06 m

Co−

cohe

renc

e

k1 delta_y0 1 2 3 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

delta_y.: 0.24 m

k1 delta_y

Figure �� Comparisons of the cocoherence of the lift forces for three angles ofwind incidence� for B�D � �� with railings� medium spire exposure� Solid line��$ dotted line� '��$ dashed line� ��

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

delta_y.: 0.06 m

Co−

cohe

renc

e

k1 delta_y0 1 2 3 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

delta_y.: 0.24 m

k1 delta_y

Figure �� Comparisons of the cocoherence of the lift forces for three angles ofwind incidence� for B�D � � with railings� medium spire exposure� Solid line� ��$dotted line� '��$ dashed line� ��


Concluding remarks

A clear link between the aerodynamic admittance and the spanwise cocoherenceof the forces was observed� For example� an observed de�cit of lift for a given chordwise strip expressed by a low crosssectional admittance was compensated for byan increase in spanwise cocoherence of the forces between the neighbouring strips�Probably the parts of the waves of incident ow that do not aect a given strip willhowever aect the unsteady character of the forces next to that strip spanwise� Itappeared that this process was strongly in uenced by the Lw�B and by the deckwidth itself� the wider the deck� the larger the cocoherence due to an increased rateof formation of secondary cross ow�

The application of the strip assumption for the deck crosssections studied hastherefore some limitations� even though the simulated Lw�B ratio was representative of fullscale conditions� The threedimensionality of the ow� and of the bueting wind loading� thus plays an important role� A �D empirical model describingthe spatial distribution of the gust loading on bridge decks is then necessary� Thisis the object of Part IIB of this research�

0.01 0.1 10.001

0.01

0.1

1

fB/V

3−D

Adm

ittan

ce

AAF= 1.0, coh=wind

AAF= Sears, coh=wind

AAF= meas., coh=forces

Lift forces, B/D= 10:1

fB/VError margin of 3−D AAF

(calc.−meas.)/meas.

AAF=Sears AAF=1

0.04 +122 % +300 %

0.08 +15 % +200 %

0.1 −25 % +124 %

0.2 −55 % +120 %

0.3 −70 % +105 %

0.4 −78 % +97 %

0.8 −65 % +500 %

Figure �� Comparisons between theory �solid and dashed lines� and direct measurements �dotted line� of the aerodynamic admittance of a span ��D admittance�AAF x JAF� for a bridge deck with a uniform mode shape�

Error margin related to the strip assumption Fig� �� compares analyticalpredictions based on the strip assumption to direct measurements of the lift forcesfor B�D � �� The quantity compared is the product of the crosssectional admittance �AAF � and the results of integration across the span of the cocoherence of thewind velocity or the lift forces� This product de�nes the aerodynamic admittance


of a span �or a �D admittance��

Calculations of the �D admittance were made for three cases� i� where quasisteady aerodynamics is assumed �AAF � ��$ ii� where AAF is represented bySears function$ and� iii� where direct measurements of AAF and cocoherence ofthe lift forces are used� Cases i� and ii� are based on the strip assumption� Fig� ��reports the error margin of the calculations� The error margin is not negligible� andfor the range of fB�V of interest the strip assumption can lead to underestimationof the wind loading�

Acknowledgements

The �nancial support of the COWIfoundation for the experimental part of thisresearch is gratefully acknowledged�

References

%�& Larose G�L��The spanwise coherence of wind forces on streamlined bridgedecks�� in Proc� of rd Int�l Colloquium on Blu� Body Aerodynamics and Ap�

plications� Blacksburg� Virginia� USA� July ��

%�& Davenport� A�G�� King� J�P�C� and Larose� G�L� �Taut strip model tests�� inProc� of Int�l Symposium on Aerodynamics of Large Bridges� Copenhagen� Feb��

%�& Larose G�L��The Response of a Suspension Bridge Deck to Turbulent Wind�

The Taut Strip Model approach� M� E� Sc� Thesis� University of Western Ontario� Canada� March ��

%�& Roberts J�B� and Surry D�� Coherence of grid generated Turbulence�� J� En�gineering Mechanics� ASCE� �� No� EM�� Dec� ��

%�& Mann J�� The spatial structure of neutral atmospheric surfacelayer turbulence��J� Fluid Mech� ��

%�& Mann J�� Kristensen L� and Courtney M�S�� The Great Belt Coherence Experi�


%�& Dennis J�E� and Woods D�J� New Computing Environments� Microcomputers

in Large�Scale Computing� A� Wouk �editor�� SIAM� ��

%�& Irwin H�P�A�H�� Wind Tunnel and Analytical Investigations of the Response of



%�& Engineering Sciences Data Unit� Characteristics of Atmospheric Turbulence

Near The Ground� Part III� variations in space and time for strong winds�Data Item �� October ��

%��& Fichtl G�H� and McVehil G�E�� Longitudinal and lateral spectra of turbulencein the atmospheric boundary layer at the Kennedy Space Center�� J� Ap�

plied Meteorology ��

%��& Fung Y�C�� An Introduction to the Theory of Aeroelasticity� Dover PublicationsInc�� N�Y��


Appendix A Photographs of the experimental set�up

Figure �� Views of the B�D�� section model in DMI�s Wind Tunnel II�


�


Figure �� Views of the B�D�� top� and B�D�� section models�


�


Figure �� Details of the centre portion of the B�D�� section model�


�


Figure �� View of the grid exposure �top� and the large spire exposure�


�

Part II�B Gust loading on bridge decks ��

Part II

Experiments and Analysis

Part II�B

Gust loading on streamlined bridge decks

Guy L� Larosea�b and Jakob Mannc


mark� �� Lyngby� Denmarkb Danish Maritime Institute� Hjortek�rsvej � �� Lyngby� Denmarkc Risoe National Laboratory� �� Roskilde� Denmark

Published in J� of Fluids and Structures ��

Abstract

The current analytical description of the bueting action of wind on longspanbridges is based on the strip assumption� However� recent experiments on closedbox girder bridge decks have shown that this assumption is not valid and is thesource of an important part of the error margin of the analytical prediction methods�In this paper� an analytical model that departs from the strip assumption is usedto describe the gust loading on a thin airfoil� A parallel is drawn between theanalytical model and direct measurements of gust loading on motionless closed boxgirder bridge decks� Empirical models of aerodynamic admittance and spanwisecoherence of the aerodynamic forces are proposed for a family of deck crosssections�

� Introduction

In the theory of bueting of slender linelike structures� Davenport clearly statedthe assumptions made as well as the limitations of the theory� One assumptionconcerns the size of the structure in relation to the size of the gusts %�&�

�� G�L� Larose and J� Mann

��that the structures �or structural members� are su�ciently slender for the

secondary span�wise �ow and redistribution of pressures to be neglected� such

that the pressures on any section of the span are only due to the wind incident

on that section� ��

This referred to the strip assumption which was also used by Liepmann as asolution to the bueting problem of aircraft wings %�&� Davenport commented thatthis assumption seemed reasonable for a thin cable or an open lattice truss but wasnot likely to be valid for structure with larger aspect ratio such as buildings %�&�

Experience has shown that this assumption appeared to be valid for the case ofbueting drag forces on slender structures where the characteristic length of thebody could be more than �� times smaller than the scales of the longitudinalcomponents of the turbulence �� m� %�&� However for the bueting liftforces� where the characteristic length of the structure can be the width of a closedbox girder bridge deck� � �� m� and the length scales of the vertical gustscould be of the order of � �� m� the assumption failed as originally expected byDavenport�

The nonvalidity of the strip assumption for the lift and torsion bueting analysishas been shown experimentally in many occasions� �rstly by Nettleton on an airfoilin grid turbulence %�&� than by Melbourne in full scale and model scale on the WestGate Bridge %�& and more recently by Larose %�� &� Sankaran et al� %�&� Kimura et al�%�&� and Bugonovic Jakobsen %��&� Larose et al� %��& �see also Part IIA� evaluatedthe error margin of the variance of the bueting response associated with the useof the strip assumption� For lift� it varied between a �� ( underestimation for areduced velocity of � to a ��( overestimation for a reduced velocity of ��

The strip assumption is nevertheless at the base of the current bueting theoryof linelike structures� both for the frequency domain and time domain approaches�

The objective is this paper is to propose the use of a model of the bueting windloads on closedbox girder bridge decks that departs from the strip assumption andreduces the error margin of the analytical prediction methods�

� Theory� lift force induced by turbulence

In this section the simplest analytical model of the lift forces induced by the gustiness of the wind acting on a motionless bridge deck is discussed� Only the verticalgusts �w component of the wind uctuations� are considered and it is assumed thatthe deck crosssection is slender enough to have aerodynamic characteristics similarto a thin airfoil � at plate��

The two main assumptions that were put forward to bring an approximate solution to the problem are� i� quasisteady aerodynamics and ii� the strip assumption�The quasisteady assumption implies that the lift force at a position y at the time


t on a bridge deck is equal to the force that would have been if the instantaneousvelocity w�t� y� persisted for an in�nitely long time everywhere on the deck�

Under the strip assumption the lift force �per unit length� on a strip at a positiony along the bridge deck is equal to the lift force per unit length that would havebeen if the wind uctuations had been fully correlated along the deck� The forceson a strip are thus assumed to be induced by the gust acting on that strip onlywithout considering the gusts on the neighbouring strips�

For an airfoil� the applicability of Sears� analysis of the unsteady lift forces inducedby a transversally fully coherent sinusoidal gust %�& is based on the strip assumption�

�� Quasi�steady aerodynamics

For a at plate with a lift slope C �z of �� the quasisteady assumption implies

Fz�x� y� � ��w�x� y��

where the lift Fz has been normalized suitably and instead of t the coordinate inthe ow direction� x � �V t is used � �V is the mean wind speed� see Fig� �� Thecovariance function of the lift force is de�ned� assuming hFzi � �� as

RL�x� y� � Fz�x

�� y��Fz�x� ' x� y� ' y�

!� ��

where h i denotes ensemble averaging� The one and twodimensional spectra of thelift force are de�ned as

SL�k��

��

Z�

��

RL�x� �� exp ��ik�x� dx ��

and

SL�k�� k��

��

Z�

��

Z�

��

RL�x� y� exp ��i�k�x' k�y�� dxdy� ��

and similarly for the vertical wind speed w�

Fourier transforming the covariance function of both sides of �� in the x andydirections� the twodimensional lift force spectrum is obtained�

SL�k�� k�� Sw�k�� k��

Equation �� can be reduced to a onedimensional force spectrum by�

SL�k��

Z�

��

SL�k�� k�� dk� ��

� ��Sw�k��


"y

x

yz

V

V

Figure �� The coordinate system has the xaxis in the direction of the mean windV � y along the bridge deck� and z perpendicular to the two other axes� V is aninstantaneous wind velocity deviating randomly from V and "y is the separationbetween strips equipped with pressure transducers�

The crossspectrum of the lift forces between strips at point � and point � separated by "y� SL�L��k��"y� � can also be derived from the twodimensional spectrum�

SL�L��k��"y� �

Z�

��

SL�k�� k��eik�ydk� ��

The spanwise root coherence of the lift forces� coh��L � is obtained by dividing

�Note that the symbol SL has several di�erent meanings� SLk�� is the force spectrum measuredon one strip of the innitely long airfoil� SL�L�k��y� is the cross�spectrum calculated frommeasurement at two strips separated by �y� and SLk�� k�� is the two�dimensional force spectrum�This could� in principle� be evaluated by measuring forces at any point along the foil and in time�and Fourier transforming both in t and y�


the crossspectrum by the onedimensional force spectrum�

coh��L �k��"y� � SL�L��k��"y�

SL�k��

�� Introducing unsteadiness

Faced with the problem of the unsteady nature of the lift forces due to wind gusts�Sears investigated the forces on an airfoil due to a special vertical velocity spectrum�in %�&��

Sw�k�� k�� Sw�k��k��

where � is Dirac�s delta function� that is� gusts with sinusoidal variations of w onlyin the direction of the mean ow �x� and no variations in the y direction or or fullycorrelated gusts�

Sears found from the linearized equations of uid motion and the KuttaJoukowskicondition �no singularities at the rear end of the airfoil� that�

SL�k�� j�� k��j�Sw�k��

where k� � k�B�� B is the deck width �or the chord of the airfoil� and�

j�� k��j� �

"""""J�� k��K��i k�� ' iJ�� k��K��i k��

K��i k�� 'K��i k��

"""""�

�Sears� function��

J�� J� are Bessel functions and K��K� are modi�ed Bessel functions of the secondkind�

An approximation to Sears� function was proposed by Liepmann and is of theform�

j�� k��j� � �

� ' �� k��Liepmann�s approximation��

With �� and �� it is clear that the lift forces reduce as k� increases� Theunsteady forces are also smaller that one would expect if quasisteady conditionswere applicable� Sears demonstrated that for any variation of the wavenumberabove �� i�e� for any ow uctuation with a �nite wavelength� the uctuating liftwill be less than the quasisteady value�

This reduction of lift with wavenumber is generally represented by the aerodynamic admittance� It can be seen as a measure of the eectivness of a body inextracting energy from the various frequency components of the turbulence�

The aerodynamic admittance can be de�ned as�

A�k�� SL�k��

C �z�Sw�k��

� ��


Based on this de�nition and on �� A�k�� for the quasisteady case� For athin airfoil in a fully correlated gust �which is equivalent to the strip assumption��

A�k�� j�� k��j��

Applying the strip assumption it is also possible to calculate the spanwise coherence of the forces�

coh��L �k��"y� � coh��w �k��"y��

Sears� function �or Liepmann�s approximation to Sears� function� is often used inbridge aerodynamics to represent the aerodynamic admittance� It tends to �� forf � �� Experiments in turbulent ow on section models of streamlined bridge deckshave shown that the admittance tends to be lower than �� for low frequencies and�if the bridge deck is blu� higher at high frequencies %�&� Furthermore� the spanwisecoherence of the lift forces was found to be much higher than for w in clear contrastto �� %�� &�

�� Lifting the strip assumption

The natural extension to Sears� analysis is due to Graham %��& from an originalsuggestion by Ribner %��&� Ribner proposed in �� as an alternative approachto the strip assumption method� to calculate the unsteady lift for an incident ow�eld represented by a superposition of plane sinusoidal wave motions of all orientations and wavelengths �equivalent to Sw�k�� k�� This required a twowavenumberaerodynamic admittance�

Graham %��& numerically computed the exact twowavenumber aerodynamic admittance using liftingsurface theory for an airfoil of in�nite span length due to gustswith arbitrary horizontal wavevectors �k�� k�� In his derivation� Graham used thesame linear assumption as in Sears� analysis but worked with yawed sinusoidal guststhat could vary in y as well as in x� That is to say that a chordwise strip could bepartially immersed in a gust oncoming from a certain horizontal direction but alsobe in uenced by a gust from a dierent direction� This is departing from the stripassumption and it is a better representation of the physics of incident gusts beingdistorted and diverted as they approach the body� An experimental validation ofGraham�s approach is presented by Jackson et al� %��&�

The twowavenumber spectrum of the lift forces becomes�

SL�k�� k�� j�� k�� k��j�Sw�k�� k��

where

j�� k�� k��j� � �

� ' �� k�

k�� ' ��

k�� ' k�� ' ��

��Mugridge��


Equation �� is an approximate closedform expression due to Mugridge %��& ofthe aerodynamic admittance� It is presented as a correction factor to the Sears�function� It reduces to �� for fully coherent gusts �when k� � ��

Other approximate closedform expressions of Graham�s exact solution exist�namely by Filotas and by Blake %��&� however� Mugridge�s approximation is mostvalid for the lower wavenumber range �k� � �� which is the range of interest forbridge aerodynamics�

Application of the twowavenumber model� Using �� it can be shown thatA�k�� do not tend to �� as k� � � %��&� This is in agreement� at least qualitativelywith results of experiments on thin airfoils and bridge decks�

The spanwise coherence is linked to the crosssectional admittance through equations �� and �� Mann %��& has calculated the crossspectrum and the spanwisecoherence of the lift forces on a thin airfoil in isotropic turbulence based on ��

and Mugridge�s approximation� It resulted that coh��L �� coh��w �coh

��L was found

larger� con�rming several experimental observations of the nonvalidity of the stripassumption� Section �� presents similar calculations�

� Experimental validations

All the quantities of section �� can be measured directly or evaluate indirectly incontrolled experiments on section models in a wind tunnel� The validation of thetwowavenumber model �referred to later on as the �D model� of the lift force isthus possible�

The ideal case would have been to conduct a series of experiments in isotropicturbulence on a thin airfoil with a lift slope approaching �� However� the ultimateobjective of this research being the application of the �D model to bridge aerodynamics� the experiments were conducted on section models with crosssectionaldimensions and proportions typical of modern closedbox girder bridge decks�

�� The experiments

Direct measurements of the bueting forces on motionless section models in turbulent ow were conducted in the DMI�s Boundary Layer Wind TunnelII� The dynamic force measurements were based on simultaneous measurements of unsteadysurface pressures on three chordwise strips of the section models� the spanwiseseparations of the strips being adjustable� The crosssection studied were typicalclosedbox girder bridge decks as seen on Fig� ��

The parameters of the experiments were� the widthtodepth ratios B�D of thesection models� the length scales of turbulence� the presence or lack of railings and


Figure �� Crosssectional shape of the models investigated� The B�D � �� crosssection was studied in another occasion and the results are reported elsewhere %�&�

the in uence of the angle of wind incidence� A detailed account of the experimentsis given in %��&�

For each case� power spectral densities of the lift and torque coe�cients werecalculated as well as the spanwise coherence of the bueting forces�

�� Description of the incident turbulent ow �eld

Three turbulent wind �elds were used in the parametric investigations of the bueting forces� two generated by spires placed �� m upstream of the model and one bya coarse grid placed �� m upstream� The two spire sets gave similar length scalesof turbulence but dierent turbulence intensities while the position of the grid wasselected to match the turbulence intensity of one of the spire sets but giving muchsmaller length scales� The spires also gave a slight anisotropy to the ow�

The grid had an isotropic mesh of �� m with a bar size of �� m� The mediumspires� in a set of three� were �� m wide at the base� �� m at the top and �� mhigh� while the large spires were �� m at the base� �� m at the top and �� mhigh� The tests in the wind tunnel were typically conducted at a mean wind speedof �� ms��

For each wind �eld the u� v and w components of the wind were �tted with theisotropic turbulence model based on the von K�arm�an spectrum described by Mannin %��&� The spectral tensor �tted particularly well the spanwise root coherence �orcocoherence� measurements made by hotwire anemometry�

The onepoint spectrum of the wcomponent was �tted with�


Table �� Characteristics of ow conditions for the three exposures in relation to thewidth of the models based on the least square �t of the von K�arm�an spectrum�

w�component Lw�B

Exposure Iw L ��

� Lw B�D � � B�D � �� B�D � ��

�� m� �m��

s� �m�

Large spires �� Medium spires �� Grid � ��

Sw�k��

��

��L�� ' �k��L�� ' k��

� ��

� ��

where L is a length scale and �� represents the rate of dissipation of turbulent

kinetic energy� Both quantity were obtained by least square �t of the measuredspectra� L was converted to L� the wavelength associated with the peak of thespectrum in its k�Sw�k�� representation� using�

Lw ��

�� ' �p��

��

L � ��L� ��

L was prefered to the conventional integral length scale since it represents thewavelength of the gusts that carry most of the energy of the turbulence�

Examples of �ts using �� are shown on Fig� �� The resulting ratios between thelength scales and the deck widths studied� Lw�B� are given in Table ��

�� Measured aerodynamic admittance functions

The aerodynamic admittances were calculated based on the experimental determination of the spectrum of the lift coe�cient SCz�f�� the lift slope and the spectra ofthe u and w component of the oncoming wind uctuations� The following expressionwas used�

jAz �f� j� ��V � SCz �f�

�C�z Su�f� ' C �

z� Sw �f�

��

The contribution of the ucomponent can be considered negligible for the crosssections studied here� as can be evaluated based on the static coe�cients of Table ��

Selected results of the direct measurements of the aerodynamic admittance arepresented in Figs� � to �� Fig� � shows the variations of the admittance for a �xedLw and varying B� On Fig� �� the admittance curves are given for a constant Bwhile the exposures are varied� It can be observed that the larger the chordtodepth ratio or the smaller the gusts� the closer are the curves to the linear theory


0.1 0.5 1 3 5 10 5010−2

10−1

k1

k1 S

_w(k

1)

large spires

0.1 0.5 1 3 5 10 5010−2

10−1

k1

medium spires

0.1 0.5 1 3 5 10 5010−2

10−1

k1

grid

Figure �� Spectra of the vertical wind uctuations for the three exposures of thisstudy� Solid line� �t of von K�arm�an spectrum using ��

for a thin airfoil �dashed lines on the graphs�� This was expected since the lineartheory is for a fully attached ow and since the larger the chrodtodepth ratio�the smaller the separation region in relation to the deck width� approaching fullyattached conditions� Also� the smaller the gusts� the smaller the length of theseparation bubble which favors reattachment of the ow�

Fig� � shows the variations of the lift aerodynamic admittance for a selection ofLw�B ratios� The same test cases are presented in Fig� � but the ordinate hasbeen multiplied by the square of the lift slope to facilitate the interpretation of theresults� All curves were found below the linear theory and an important de�cit of liftwas found in the low reduced frequency range� Even though the deck crosssectionstudied had poorer lift characteristics than a thin airfoil� they could generate hasmuch lift than the linear theory predicted at higher reduced frequencies for the sameinput wind spectrum� The bodyinduced turbulence is probably the source of theobserved larger than anticipated lift�


Table �� Time averaged aerodynamic force coe�cients �based on B� for the mediumspire exposure� deck without railings�

B�D Cz Cm C �z C �

m

� ��

0.001 0.01 0.1 10.01

0.1

1

fB/V

Aer

odyn

amic

adm

ittan

ce

without railings, lift

medium spires

B/D= 5:1

B/D= 10:1

B/D= 12.67:1

0.001 0.01 0.1 10.01

0.1

1

fB/V

without railings, lift

grid

B/D= 5:1

B/D= 10:1

B/D= 12.67:1

Figure �� Variations of the aerodynamic admittance with reduced frequency forthe medium spire and the grid exposures for three B�D ratios� The thick line isLiepmann�s approximation �� to Sears� function�


without railings

with railings

0.001 0.01 0.1 10.01

0.1

1

Aer

odyn

amic

adm

ittan

ce

B/D=10:1

large spires

med. spires

grid

0.001 0.01 0.1 10.01

0.1

1

fB/V

Aer

odyn

amic

adm

ittan

ce

B/D=12.67:1

large spires

med. spires

grid

0.001 0.01 0.1 10.01

0.1

1

B/D=5:1

Aer

odyn

amic

adm

ittan

ce

med. spires

grid

Figure �� Variations of the lift aerodynamic admittance with reduced frequency forall exposures and all B�D�


0.001 0.01 0.1 10.01

0.1

1

fB/V

Aer

odyn

amic

adm

ittan

ce

Lift, without railings

[1.50]

[0.73]

[0.58][0.57]

[0.29]

[0.23]

Figure �� Variations of the vertical aerodynamic admittance with reduced frequency�The numbers in % & refer to the Lw�B ratio�

0.001 0.01 0.1 1 3

1

10

fB/V

Aer

odyn

. adm

ittan

ce x

Cz´

*2


[1.50]

[0.73]

[0.58][0.57]

[0.29]

[0.23]

Figure �� Lift aerodynamic admittance functions for six dierent Lw�B ratio� Theadmittance has been multiplied by C �

z� to be compared with Liepmann�s approxi

mation to Sears� function for a at plate with a lift slope of �� dash line��


�� Measured spanwise cocoherence of the forces

The experimentally determined spanwise distribution of the lift forces are presented here under the form of cocoherence function or normalized cospectrum�Cocoherence is the real part of the crossspectrum of the forces between two stripsnormalized by the onepoint spectrum� The advantage of the cocoherence presentation over the root coherence is that it can show if the forces become negativelycorrelated for higher frequencies or large separations�

The spanwise cocoherence of the lift forces are presented on Figs� � and �� Thetest conditions selected are for the medium spire exposure and the B�D � � and ��ratios respectively� Also shown on the plots are the variations of the cocoherenceof the w component of the wind for the medium spire exposure� The curves shown�solid line� are based on a �t of the measured cocoherence data with the vonK�arm�an spectral tensor for isotropic turbulence described in %��&� The cocoherenceof the forces was found larger the the cocoherence of the incident wind�

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.03 m

Co−

cohe

renc

e

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1delta_y.: 0.13 m

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1delta_y.: 0.24 m

k1 delta_y

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1delta_y: 0.3 m

k1 delta_y

Figure �� Variations of spanwise cocoherence as a function of k�"y for the liftforces �dotted line� for Lw�B � �� compared to the wind uctuations �solid line�w with L� ��


0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.03 m

Co−

cohe

renc

e

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1delta_y.: 0.13 m

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1delta_y.: 0.24 m

k1 delta_y

Co−

cohe

renc

e

0 1 2 3 4

0

0.5

1delta_y: 0.3 m

k1 delta_y

Figure �� Variations of spanwise cocoherence as a function of k�"y for the liftforces �dotted line� for Lw�B � �� compared to the wind uctuations �solid line�w with L� ��

�� Applications of the ��D analytical model

The twowavenumber spectra Sw�k�� k�� modelling the ow �elds of the experimentswas calculated using the von K�arm�an isotropic turbulence tensor %��& and the �ttedlength scales given in Table ��

Calculations of j A�k�� j and coh��L �k��"y� using the �D model of Section ��

and Mugridge�s approximation �� were made for a series of Lw�B ratio corresponding to the experimental conditions� The results of the analytical calculationsare presented on Figs� �� and ��


0.001 0.01 0.1 10.01

0.1

1

fB/V

Aer

odyn

amic

adm

ittan

ce

[1.50]

[0.73]

[0.58]

[0.29]

[0.23]

Liepmann´s approx.

Figure �� Analytical calculation of the aerodynamic admittance for a thin airfoilusing �� and Mugridge�s approximation� for isotropic turbulence with length scaleLw for dierent values of B� The numbers in % & refer to the Lw�B ratio�

0 0.5 1 1.5 2

0

0.5

1

Co−

cohe

renc

e delta_y.: 0.03 m

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1

Co−

cohe

renc

e delta_y.: 0.13 m

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1

Co−

cohe

renc

e

k1 delta_y

delta_y.: 0.24 m

0 1 2 3 4

0

0.5

1

k1 delta_y

delta_y.: 0.3 m

Figure �� Analytical calculations of spanwise cocoherence of lift forces based onthe �D model and Mugridge�s approximation� Thin line� Lw�B � ��$ dashed line�Lw�B � �� and thick solid line� wcomponent of the wind uctuations using thevon K�arm�an spectral tensor with L��


�� Discussions

A qualitative agreement was observed between the measured admittance and theanalytically calculated admittance using the �D model� The agreement improvesas the widthtodepth ratio increases or as the aerodynamics of the crosssectionsapproach the aerodynamics of a at plate� As observed earlier %�&� the bluer thebody� the larger the admittance at higher frequency compared to the prediction fora thin airfoil�

Another important qualitative agreement refers to an admittance value dierentfrom �� for low reduced frequency� For the larger Lw�B ratios� the analyticalcalculations did not give an admittance value larger than �� even at low reducedvelocity� Also� calculated and measured admittance curves for low Lw�B ratio havesimilar shapes� Going from low frequency to higher frequency the curves have atrough corresponding to the largest value of Sw� then a peak corresponding to thelargest value of SL and �nally have a sudden drop with more or less the same slope�This apparent point of in ection in the admittance curves can be observed for caseswhere the incident vortices are small in comparison to the deck width� resulting ina frequency shift between the peak of the wind spectrum and the peak of lift forcespectrum�

0.001 0.01 0.1 10.01

0.1

1

fB/V

Aer

odyn

amic

adm

ittan

ce

a)

[1.50]

[0.58]

[0.23]

0.001 0.01 0.1 1

1

10

fB/V

Aer

odyn

. adm

it. x

Cz´

*2

b)

[1.50]

[0.58]

[0.23]

Figure �� Comparisons between analytical calculations of lift aerodynamic admittance using the �D model for a thin airfoil and direct measurements on bridgedecks for three Lw�B ratios� The dashed line is Liepmann�s approximation to Sears�function� On b� the ordinate has been multiplied by the square of the lift slope�

Figs� �� and �� show quantitative comparisons of the analytical calculations anddirect measurements of admittance for selected Lw�B ratios� On Fig� �� a� it canbe seen that the analytical calculation approaches the measured admittances at very


0.001 0.01 0.1 1

1

10

fB/V

Aer

odyn

amic

adm

ittan

ce x

Cz´

*2 Liepmann´s approx.

3−D model

without railings

with railings

Figure �� Comparisons between analytical calculation �thick solid line� of lift aerodynamic admittance using the �D model for a thin airfoil and direct measurementon a bridge deck with B�D�� for Lw�B � �� The ordinate has been multipliedby the square of the lift slope�

low reduced frequency� This aspect is positive since it indicates that for quasisteadyconditions� the linear �D theory and the experiments are in agreement� It alsovalidates some previously publised measurements of the aerodynamic admittanceof blu bodies that have been disregarded since they did not tend to �� for verylow reduced frequency� For higher fB�V � the bridge decks produce bodyinducedturbulence that increases their admittance characteristics in comparisons with thetheory�

Fig� �� b� compares the product of Az�f� and C �z� for the theory and the exper

iments� It indicates that if the test case approach fully reattached ow conditions�the bridge deck could generate similar lift than a thin arifoil� This aspect is alsoillustrated on Fig� �� where the case with railings� which has shown a smaller separation bubble� has lift characteristics closer to the �D theory than the case withoutrailings�

Spanwise cocoherence� Qualitative agreement was also observed between analytical calculations with the �D model and direct measurements of the spanwise

cocoherence of the forces� Most importantly coh��L was found larger than coh

��w

con�rming earlier observations� The calculations also reinforced the observation reported in %��& that the lower the aerodynamic admittance the larger the spanwisecocoherence con�rming the link between the two quantities�


0 0.5 1 1.5 2

0

0.5

1

Co−

cohe

renc

e delta_y.: 0.03 m

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1

Co−

cohe

renc

e delta_y.: 0.13 m

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1

Co−

cohe

renc

e

k1 delta_y

delta_y.: 0.24 m

0 1 2 3 4

0

0.5

1

k1 delta_y

delta_y.: 0.3 m

Figure �� Comparisons between calculations �solid line� and direct measurements�dotted line� of the spanwise cocoherence of the lift forces as a function of k�"y for

Lw�B � �� The dashed line represents the variations of cocoh��w of the incident

wind �L�� m��

Direct comparisons between calculations with the �D model and experiments areshown on Figs� �� and �� The agreement is excellent for the smaller separations inthe lower wave number range� For the smallest separation� �� m� the calculationoverestimated the cocoherence for all k�"y� In fact� for this case� cocohL �cocohw� This observation� for very small separation in comparison to the deckwidth� was also reported in other studies %�� & and can not really be explained bythe theory presented here�

It can be observed also by comparing Fig� �� to Fig� �� that as Lw�B reducesor as the in uence of the small scale turbulence increases� the �D model underestimated the large cocoherence measured at higher wave numbers during the experiments� The ow mechanisms involved in the reattaching shear layer �as describedin Part �A� which has shown a tendency to spread spanwise the vortices at highfrequencies can explained the dierence� since the theory does not take this eectinto account� This underestimation however is compensated by a higher aerodynamic admittance obtained from the calculations when compared to the theory�


0 0.5 1 1.5 2

0

0.5

1

Co−

cohe

renc

e delta_y.: 0.03 m

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

0 1 2 3 4

0

0.5

1

Co−

cohe

renc

e delta_y.: 0.13 m

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

0 1 2 3 4

0

0.5

1

Co−

cohe

renc

e

k1 delta_y

delta_y.: 0.24 m

0 1 2 3 4

0

0.5

1

k1 delta_y

delta_y.: 0.3 m

Figure �� Comparisons between calculations �solid line� and direct measurements�dotted line� of the spanwise cocoherence of the lift forces as a function of k�"y for

Lw�B � �� The dashed line represents the variations of cocoh��w of the incident

wind �L�� m��

Pressure distribution� The earlier remark concerning the measurement of largerAz�k�� than calculations for the higher frequencies for the bluer bridge deck canpartly be explained by inspection of the unsteady pressure distribution�

Figs� �� and �� present mean and rms pressure distribution for the medium spireexposure for B�D � � and B�D � �� cases respectively� The pressure uctuationsfor the �� case are concentrated near the leading edge while the uctuations forthe �� case are extended more downstream� almost up to midchord� The latterrepresent probably a shedding zone of vortices at or just after the reattachmentpoint� These vortices will later on mix with the wake and in uence the formationof lift� It is the zone of formation of what has been called the signature turbulence

or the bodyinduced turbulence� For the �� case� bodyinduced turbulence is alsopresent but it is believed that it does not mix in the same manner with the wakesince the convection length and the velocity is greater� The vortices are likely tobe distorted by the mean velocity �eld enveloping the body following the rapiddistortion theory� The rms pressure distribution for the �� and �� modelsapproaches the pressure distribution for the thin airfoil case�




95146jm0.026

Flow

Figure �� Surface pressure distribution on B�D� �� medium spire case� withoutrailings� �� ms�� Scaling� depth of deck� mean Cp of � and rms Cp of ��Polarity� negative away from the surface� The rms distribution is given for theprojected tributary area�



95146jm0.235

Flow

Figure �� Surface pressure distribution on B�D� �� large spire exposure� withoutrailings� �� ms�� Scaling� depth of deck� mean Cp of � and rms Cp of ��


The suggestion that the bodyinduced turbulence is the cause of the higher liftadmittance for bridge decks than the theory predicts for a thin airfoil also appliesto the torsional admittance� In fact for torsion� this eect should be even greatersince the zones that are aected on the deck �at the trailing edge� have an importantcontribution to the torsional forces� This is con�rmed on Fig� �� where the measuredaerodynamic admittance in torsion � is compared to Liepmann�s approximation forthe lift admittance�

0.001 0.01 0.1 10.01

0.1

1

Aer

odyn

amic

adm

ittan

ce

fB/V

[1.50]

[0.73]

[0.58][0.57]

[0.29]

[0.23]

Torsion, without railings

Figure �� Variations of the torsional aerodynamic admittance with reduced frequency� The numbers in % & refer to the Lw�B ratio� The thick line shows theLiepmann�s approximation to the lift admittance�

�The experimental determination of the torsional admittance is similar to the lift admittance�The subscript z in equation �� is replaced by subscript m�


� Empirical model of the spatial distribution of the

bu�eting forces on closed�box girder bridge decks

The �D analytical model of the gust loading on a thin airfoil has shown qualitativeagreement with the experimental description of the spatial distribution of the windforces on bridge decks� It has helped to understand the generation of lift on achordwise strip and has explained the observed larger spanwise coherence of theaerodynamic forces�

Quantitatively however� it has failed to reproduce closely the aerodynamic admittance of the bluer closedbox girder bridge decks in the �� to � reduced frequencyrange that is of interest for longspan bridge aerodynamics� For the more slenderbridge decks� e�g� B�D � �� it is believed that it could be used as an alternativeto the Sears� function�

In the following� an empirical model of the spatial distribution of the wind forcesdue to turbulence on closedbox girder bridge decks is presented� The model isinspired by the �D analytical model and is based on direct measurements of thegust loading�

�� Cross�sectional aerodynamic admittance as a function of Lw�B

By inspection of the aerodynamic admittance curves� either obtained from analyticalcalculations or direct measurements� a quasilinear dependence on the Lw�B ratiowas depicted� To quantify this dependence� attempts to collapse the admittancecurves on one line were made� The best collapses are shown in Fig� �� both forthe analytical calculations and the experiments� The similarities between the setsof curves are remarkable� The best collapse was obtained when the ordinate wasnormalized by Lw�B to some power and when fB�V was further reduced also byLw�B to some power�

Based on this collapse� an empirical expression that could describe the aerodynamic admittance as a function of Lw�B and fr � fB�V was sought� The best �tof the measured aerodynamic admittance curves was obtained with the followingexpression�

""""Az�m�fr�Lw

B�

""""� �a �� '

pfr�

� ' b f�r� ��

where

a � ��

�Lw

B

��and b � ��

�Lw

B

��for lift$ and� ��

a � ��

�Lw

B

��and b � ��

�Lw

B

��for moment� ��


0.001 0.01 0.1 10.01

0.1

1

fB/V x L_w/B

Nor

mal

ized

aer

odyn

amic

adm

ittan

ce

a) 3−D analytical model

0.001 0.01 0.1 10.01

0.1

1

fB/V x (L_w/B)^(2/5)

b) experiments

Figure �� Collapse of lift aerodynamic admittance functions for �ve Lw�B ratios�varying from �� to �� In a� the admittances calculated using the �D modelhave been normalized by �Lw�B�� and are plotted versus �fB�V � �Lw�B�� Inb� the measured admittances have been normalized by �Lw�B�� and are plottedversus �fB�V � �Lw�B��

The limits of validity of the empirical expression above� related to the family ofclosedbox girder bridge decks studied here� are�

� � B�D � �� and �� Lw�B � ��

Examples of the �t of equations �� to the experimental data are given in Figs� ��and ��

�� Model of the span�wise co�coherence of the aerodynamic forces

Integral length scales� The width of the correlation has been represented inPart IB by the integral length scales�

Ly �

Z�

�R��"y�d�"y� ��

where R�� is the correlation coe�cient� The correlation width can be calculatedfrom the measured correlation coe�ent of the input� the w component� and directlycompared to the correlation width of the output� the aerodynamic forces� On thebasis of the strip assumption� for very large gusts compared to the size of thestructure� Ly

L�M should tend to Lyw for a motionless structure�

The correlation width of the aerodynamic forces is compared in Fig� �� to thecorrelation width of the w component for all the test cases of this study� Alsoincluded in the graphs are results obtained by other researchers �see Part IB� forsimilar streamlined bodies but for dierent experimental conditions�


0.001 0.01 0.1 10.01

0.1

1

fB/V

Aer

odyn

amic

adm

ittan

ce


[1.50]

[0.73]

[0.58]

[0.29]

[0.23]

Figure �� Variations of lift aerodynamic admittance functions for �ve Lw�B ratios�numbers in % &�� The smooth solid lines show a �t of the measured data usingequation ��

0.001 0.01 0.1 10.01

0.1

1

fB/V

Aer

odyn

amic

adm

ittan

ce

Moment, without railings

[1.50]

[0.73]

[0.58]

[0.29]

[0.23]

Figure �� Variations of moment aerodynamic admittance functions for �ve Lw�Bratios �numbers in % &�� The smooth solid lines show a �t of the measured data usingequation ��


The data points appeard to fall well on one line when plotted as a function ofLw�B and tends toward unity as Lw�B increases as assumed by the strip theory�From Fig� �� the width of in uence of the gusts can be determined for a givenLw�B ratio� For conditions representative of fullscale conditions for closedboxgirder bridge decks �� Lw�B � �� the width of the in uence of the vertical gustson the lift forces was found to be at least � times the integral length scale of thegusts themselves�

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21

2

3

4

5

6

yL_L

/ yL

_w

Lift

B/D=5 B/D=10 B/D=12.67

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21

2

3

4

5

6

L_w/B

yL_M

/ yL

_w

Moment

B/D=5 B/D=10 B/D=12.67

Figure �� Variations of the ratio of the integral length scales of the forces tothe integral length scales of w� Ly

L�M�Lyw� as a function of the Lw�B ratio� All

data points are from the present research� with the exception of� � Nettleton %�&$�� Larose %�&$ and �� Bogunovic Jakobsen %��&�

Force cocoherence� By inspection of the variations of the cocoherence of thelift forces with k�"y it was observed that�

�� the shape and magnitude of the curves have more a�nity with the cocoherenceof v than w� This a�nity is though to be only coincidental$

�� the decay of the force cocoherence is slower than the incident wind cocoherence and appears to be a function of the spanwise separation�

In search for an adequate empirical model of the force cocoherence curves� attempts where made to �t the data with the analytical expressions of the cocoherence


of the wind derived from the von K�arm�an spectrum in Mann et al� %��&� The expressions for the v and w components are reproduced here�

cocohv�"y� k��

#��

� ��

�

� ��

K�� '

��"yk��

�� ' ��"yk��K��

��

cocohw�"y� k��

#��

� ��

�

� ��

��K��

��yk�

�� ' ��"yk��

�K��

��

where K�� are modi�ed Bessel functions of the second kind and�

� �

�"y�k�� '

"y�

L�

� ��

� k�"y

s� '

�

k��L�

��

is the von K�arm�an collapsing parameter�

A similar analysis was made by Kimura et al� %�& for at cylinders� At �rst thecalculated cocoherence using the �D model and Mugridge�s approximation where�tted with equations �� and �� The expression for w did not �t particularlywell the lift cocoherence when compared to the �t obtained with the expressionfor v �as observed in point � above�� To improve further the �t� the wave numberk� in equation �� was raised to an exponent a as suggested by point � and asproposed by Kimura et al� The exponent a and the length scale L were kept as oating parameters for the least square �ts� An example of the �t is given in Fig� ��and the results for three Lw�B ratios are shown in Fig� ��

The best �t was obtained when a� �� and when LL� the length scale relatedto the lift forces� was varied as a function of "y as seen on Fig� �� It was alsoobserved that if LL was normalized by L� the von K�arm�an length scale describingthe input wind �eld� and plotted versus normalized separation "y�B� the results�tted well on one line and tend to a value slightly lower than � �see righthandsidegraph of Fig� ��

In a similar fashion� all measured cocoherence of the aerodynamic forces of thisstudy were �tted with equation �� with ka� � keeping the exponent a and the lengthscale LL as a oating parameter for the least square �ts� An example of the �t isgiven in Fig� �� and the results for all Lw�B ratios are shown in Fig� �� It canbe observed from Fig� �� that the normalized lift length scale LL�L also tends toa value near �� for larger "y�B and that the curves are grouped by chordtothickness B�D� For example� the Lw�B �� and �� are from B�D�� in twodierent exposures� and showed exactly the same LL�L and a for both cases�

For separations smaller than half the deck width it is clear that the larger thedeck width� the larger the lift length scales� It indicates that the longer the incidentlargeeddies are being distorted over the deck� the larger the possibility of formingsecondary cross ow� the larger the spanwise coherence�


The exponents a also appeared to be grouped by B�D ratios$ the larger the deckwidth� the smaller the exponent up to a certain plateau� This exponent gives anindication of the rate of decay of the cocoherence with frequency� As described inPart IB� blu bodies have a tendency to spread the separated ow vortices �bodyinduced turbulence� spanwise as they are convected over the body� Since the largerthe afterbody the larger this eect� it could explain the observed relatively largerin uence of the higher frequency gusts for the wider decks� represented by smallerexponents a�

The value of a for B�D � �� and �� appeared to tend toward �� for larger"y�B� This is in agreement with results by Kimura et al� %�& who obtained a��for at cylinders with B�D � ��

It was observed that� at high frequencies� the measured cocoherences were inmost cases larger than predicted by the analytical model �see Fig� �� The failureof the �D analytical model �chaindotted line in Fig� �� and solid lines on Fig� ��has probably to do with separated ow vortices� which may have strong spanwisecocoherence� Separated ow is out of reach of the analytical model�

0 0.5 1 1.5 2

0

0.5

1

Co−

cohe

renc

e delta_y.: 0.03 m

L: 1.37 m, a: 0.95

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

L: 1.1 m, a: 0.95

0 1 2 3 4

0

0.5

1

Co−

cohe

renc

e delta_y.: 0.13 m

L: 0.7 m, a: 0.95

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

L: 0.6 m, a: 0.95

0 1 2 3 4

0

0.5

1

Co−

cohe

renc

e

k1 delta_y

delta_y.: 0.24 m

L: 0.48 m, a: 0.95

0 1 2 3 4

0

0.5

1

k1 delta_y

delta_y.: 0.3 m

L: 0.44 m, a: 0.95

Figure �� Example of the least square �t of the force cocoherence� calculated usingthe �D model and Mugridge�s approximation and �tted with equation �� withka� for Lw�B� ��


0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

delta_y (m)

L_L

(m)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

delta_y /B

L_L

/ L

L_w/B= 1.50L_w/B= 0.73L_w/B= 0.29

Figure �� Results of the �t of the cocoherence of the lift forces calculated usingthe �D model and Mugridge�s approximation and �tted with equation �� withka� �

0 0.5 1 1.5 2

0

0.5

1

Co−

cohe

renc

e

delta_y.: 0.03 m

L: 0.2356 m, a:1.199

0 0.5 1 1.5 2

0

0.5

1delta_y.: 0.06 m

L: 0.82 m, a:1.028

0 1 2 3 4

0

0.5

1

Co−

cohe

renc

e

delta_y.: 0.13 m

L: 0.5088 m, a:0.8597

0 1 2 3 4

0

0.5

1delta_y.: 0.16 m

L: 0.4548 m, a:0.8455

0 1 2 3 4

0

0.5

1

Co−

cohe

renc

e

k1 delta_y

delta_y.: 0.24 m

L: 0.414 m, a:0.8243

0 1 2 3 4

0

0.5

1

k1 delta_y

delta_y.: 0.3 m

L: 0.3782 m, a:0.8123

Figure �� Example of the least square �t of the measured cocoherence of the liftforce with equation �� with ka� for Lw�B� ��


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

Nor

mal

ized

lift

leng

th s

cale

solid

B/D=5

+=[1.5]

o=[0.57]

dashed

B/D=10

+=[0.80]

o=[0.73]

x=[0.29]

dotted

B/D=12.67

+=[0.63]

o=[0.58]

x=[0.23]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.6

0.7

0.8

0.9

1

1.1

delta_y / B

Wav

e nu

mbe

r ex

pone

nt

Figure �� Results of the �t of the measured cocoherence of the lift forces asa function of "y�B� The ordinate of the top graph is the lift length scale LL

obtained from the �t of equation �� and normalized by the length scale L of theincident ow� The chaindotted line refers to the theoretical calculations with the�D model� The wave number exponent refers to a in ka� �


Empirical formulations

Relationships between LL� a and "y�B were extracted from the results presentedabove� In combination with the analytical description of the incident wind �eld� theyform the basis of an empirical model of the spatial distribution of the aerodynamicforces on closed box girder bridge decks of the crosssection family studied here�

The spanwise cocoherence of the lift forces can be expressed as a function of"y� k��

Lw

B � BD by�

cocohL��

#��

� ��

�

� ��

K�� '

��"yka� ��

�� ' ��"yka� ��K��

��

where� ��

� � ka�"y

s� '

�

�ka�LL��

a �

�B

D

�� p' yB ��

�q ' r yB ��

$ p � �� q � �� r � ��

LL � L�p' y

B ��

�q ' r yB ��

$ p � �� q � �� r � ��

A graph showing the variations of LL and a as a function of "y�B for the liftforces is given in Fig� �� For the sake of simplicity� the variations of LL�L with"y extracted from the calculations of cocoherence with the �D model were usedto represent LL in equation �� The eect of this simpli�cation was found to benegligible and is illustrated in the following�

Fig� �� shows � graphs of the cocoherence as a function of � for all Lw�B ratiosof this study� for all separations� The cocoherence data collapsed very well� In a�the measured cocoherence is plotted versus � calculated directly using the resultsof the �tting of LL and a as reported in Fig� �� The dark solid line represents thebest �t of the collapsed data with�

cocohL�k��"y�Lw

B�B

D� � exp%�c��c� & cos�c��

where c�� c� and c� are given in Fig� �� This expression was used in Part IIA tomodel the cocoherence of the incident wind �eld�

In b�� the measured cocoherences are also plotted versus � but with LL and a

calculated using equations �� and �� The resulting curve is practically identicalto a� con�rming that the simpli�cation for LL is adequate� In c� the cocoherencecalculated using the empirical model� equations �� to �� is plotted as a functionof �� The resulting c�� c� and c� describe the same function as in a� and b� of Fig� ��


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

Nor

mal

ized

lift

leng

th s

cale

p= 1.004; q= 0.4582; r= 1.42

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21

1.5

2

2.5

3

delta_y / B

a x

(B/D

)^0.

25

p= 0.1599; q= 0.08787; r= 0.9348

Figure �� Collapse of LL and a based on the �t of the cocoherence of the liftforces with equation �� with ka� for all cases�

The cocoherence calculated with the �D analytical model were also plotted asa function of �� The collapsed data �t well equation �� with� c�� c�� and c��

Joint acceptance function� The evaluation of the spatial distribution of theaerodynamic forces is at the hearth of the calculation the joint acceptance function�JAF � As de�ned in Part IB� this quantity is a measure of the correlation betweenthe vibration mode and the wind loading across the span�

To assess the adequacy of the empirical model described above� the joint acceptance function was calculated for a uniform mode shape� ��x�� over a �� mspan �section model span�� for B�D� �� L�� m� The calculations are reportedin Fig� �� as a function of reduced frequency for � conditions�

�� on the basis of the strip assumption� cocohL � cocohw$

�� using the �t of the calculated cocoherence of the forces for a thin airfoil withthe �D model and Mugridge�s approximation$

�� using the directly measured cocoherence of the lift forces$

�� and using the empirical model described above �� with the �tted LL and a�


0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1C

o−co

here

nce

measurements with meas. a and meas. L

c1,c2,c3: 0.3462, 1.496, 0

0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1

Co−

cohe

renc

e

measurements with fitted a and theo. L

c1,c2,c3: 0.3203, 1.519, 0

0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1model with fitted a and theo. L

eta

Co−

cohe

renc

e

c1,c2,c3: 0.3815, 1.351, 0

Figure �� Variations of the measured cocoherence of the lift forces as a functionof � for all separations and all Lw�B of this study� The solid line is the best �t ofthe collapsed data with equation �� where the constants c�� c� and c� are givenin the graphs�

The calculations on the basis of the strip assumption clearly underestimated thejoint acceptance function while the empirical model closely matched the JAF basedon the directly measured cocoherence of the lift� Calculations with the �D model�case �� gave a good approximation to the JAF in the lower fB�V range but failedto follow the mode shape of the JAF in the higher range �� fB�V � �� Case� has a constant wave number exponent �a�� for all spanwise separations�therefore does not decay in the same manner as the measured cocoherence for aclosed box girder bridge deck� The shape of the JAF for cases � and � are consistentwith the results reported earlier in %�& and summarised in Part IB�

A similar shape of the JAF was obtained by Hoi %��& from measurements on asection model of the BronxWhitestone Bridge� Hoi made one of the rare direct


0.001 0.01 0.1 10.01

0.1

1

fB/V

Join

t acc

epta

nce

func

tion

Wind, case 1

Forces, case 2

Forces, case 3 and 4

Figure �� Variations of the joint acceptance function with reduced frequency forfour cases�

measurements of the joint acceptance function by simultaneously measuring thetotal lift forces at the extremities of a section model in motion and the lift forceson a oating segment at the center of the model� Dividing the �rst quantity by thesecond gave a JAF curve in excellent agreement with the curve obtained here� eventhough Hoi�s experiments was for a truss girder bridge�

Cocoherence of the overturning moment� The analysis described above wasrepeated for the aerodynamic overturning moment of the bridge deck and the resultsare summarised in Figs� �� to �� Similar variations of the LM�L and a with "y�Bwere obtained when compared with the lift force results� The scatter of LM�L for"y�B � �� was lower than observed for the lift forces and coincidently followedthe variations of LL extracted from the �D analytical model �chaindotted line inFig� �� The exponent a was found to be slightly higher than for the lift indicatingslightly more rapid decay of the moment cocoherence� The best �t of the exponenta was obtained with�

a �

�B

D

�� p' yB ��

�q ' r yB ��

$ p � �� q � �� r � ��

as shown in Fig� ��


The moment cocoherence was plotted as a function of � �� with a de�ned by�� and LM�L de�ned by �� The results are shown in Fig� �� for the same casesas Fig� �� The collapse of the moment cocoherence was found to be even betterthan for the lift cocoherence and was well described by the empirical model�

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

Nor

mal

ized

lift

leng

th s

cale

solid

B/D=5

+=[1.5]

o=[0.57]

dashed

B/D=10

+=[0.80]

o=[0.73]

x=[0.29]

dotted

B/D=12.67

+=[0.63]

o=[0.58]

x=[0.23]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.6

0.7

0.8

0.9

1

1.1

delta_y / B

Wav

e nu

mbe

r ex

pone

nt

Figure �� Results of the �t of the measured cocoherence of the torsional forcesas a function of "y�B� The chaindotted line refers to the theoretical calculationswith the �D model� The wave number exponent refers to a�


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21

1.5

2

2.5

3

delta_y / B

a x

(B/D

)^0.

15

p= 0.09764; q= 0.05871; r= 0.9701

Figure �� Collapse of a based on the �t of the cocoherence of the torsional forceswith equation �� with ka� for all cases�

0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1

Co−

cohe

renc

e

measurements with meas. a and meas. L

c1,c2,c3: 0.3409, 1.333, 0.2245

0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1

Co−

cohe

renc

e

measurements with fitted a and theo. L

c1,c2,c3: 0.3102, 1.225, 0.299

0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1model with fitted a and theo. L

eta

Co−

cohe

renc

e

c1,c2,c3: 0.3813, 1.345, 0

Figure �� Variations of the measured cocoherence of the overturning moment asa function of � for all separations and all Lw�B of this study� The solid line is thebest �t of the collapsed data with equation �� where the constants c�� c� and c�are given in the graphs�


Conclusions

An analytical model that departs from the strip assumption was presented to describe the gust loading on a thin airfoil in isotropic turbulence� The �D model isbased on a two horizontal wavevector analysis as opposed to the fully correlatedsinusoidal gust analysis due to Sears�

The �D model has shown qualitative agreement with the experimental description of the aerodynamic admittance and the spatial distribution of the gust loadingon bridge decks� It has helped understand the generation of lift on a chordwisestrip and has explained the observed larger spanwise cocoherence of the forceswhen compared to the cocoherence of the incident wind�

Quantitatively however� the �D analytical model did not reproduce adequatelysome aspects of the gust loading on the bluer closed box girder bridge decks� Anempirical model was thus derived to describe the gust loading on the family of closedbox girder bridge decks of this study� The empirical model departs from the stripassumption and was inspired by the analytical description of the incident wind �eldas well as the �D model of the spatial distribution of the aerodynamic forces�

It is likely that the empirical model could be extended to the crosssections ofother closed box girder bridge decks provided that the aerodynamic characteristicsapproached the at plate characteristics� For relatively open truss girder bridgedecks the strip assumption would certainly provide a fair approximation at lowfrequencies but the bodyinduced turbulence will dominate at high frequencies�In this situation� the recommendation is to rely on ad hoc wind tunnel tests todetermine the gust loading�

Acknowledgements� The �nancial support of the COWIfoundation for the experimental part of this research is gratefully acknowledged as well as the researchscholarship provided by the Danish Research Academy�

References

�� Davenport AG� �The response of slender� line�like structures to a gusty wind�� inProc� of Institution of Civil Engineers� �� Nov ��

�� Liepmann HW��On the application of statistical concepts to the bu�eting problem��J� Aero� Science� �� no �� Dec �� s

�� Vickery BJ� �Fluctuating lift and drag of a long cylinder of square cross�section in asmooth and turbulent stream�� J� Fluid Mech� ��

�� Ektin B� Dynamics of Atmospheric Flight� John Wiley and Sons� �� pp ��

�� Melbourne W H� �Comparison of model and full�scale tests of a bridge and chimneystack�� in Proc� of Int�l Workshop on Wind Tunnel Modelling� Maryland� USA� ��


�� Larose GL�The Response of a Suspension Bridge Deck to Turbulent Wind� The Taut

Strip Model approach� MESc Thesis� University of Western Ontario� Canada� March��

�� Larose GL� Davenport AG� and King JPC� �On the unsteady aerodynamics forcesacting on a bridge deck in turbulent �ow�� in Proc� of the �th US Nat� Conf� on Wind

Engineering� UCLA� Los Angeles� USA� June ��

� � Sankaran R and Jancauskas ED� �Measurements of cross�correlation in separated�ows around blu� cylinders�� J� Wind Eng� Ind� Aerodyn��

�� Kimura K� Fujino Y� Nakato S and Tamura H� �Characteristics of the bu�etingforces on �at cylinders�� in Proc� of �rd Int�l Colloquium on Blu� Body Aerodynamics

and Applications� Blacksburg� Virginia� USA� July ��

�� Bogunovic Jakobsen J� Fluctuating wind load and response of a line�like engineering

structure with emphasis on motion�induced wind forces� PhD Thesis� The NorwegianInstitute of Technology� Trondheim� Norway� ��

�� Larose GL� Tanaka H� Gimsing NJ and Dyrbye C� �Direct measurements of buf�feting wind forces on bridge decks�� accepted for publication in Proc� of �nd European

African Conf� on Wind Eng�� Genova� Italy� June ��

�� Fung YC� An Introduction to the Theory of Aeroelasticity� Dover Publications Inc�NY� ��

�� Graham JMR� �Lifting�surface theory for the problem of an arbitrary yawed sinu�soidal gust incident on a thin aerofoil in incompressible �ow�� Aeronautical Quaterly�� May ��

�� Jackson R� Graham JMR and Maull DJ� �The lift on a wing in a turbulent �ow��Aeronautical Quaterly� �� part �� Aug ��

�� Mugridge BD� �Gust loading on a thin aerofoil�� Aeronautical Quaterly� �� Aug��

�� Blake WK� Mechanics of ow�induced sound and vibration� Volume �� AcademicPress Inc� Orlando� Florida� �� p ��

�� Mann J� �The strip assumption in bridge aerodynamics�� unpublished Technical Note�Danish Maritime Institute� Oct ��

�� Mann J� �The spatial structure of neutral atmospheric surface�layer turbulence��J� Fluid Mech� ��

�� Mann J� Kristensen L and Courtney MS� The Great Belt Coherence Experiment �

A study of atmospheric turbulence over water� Ris� Report No R�� p

�� Hoi CL� Measurements of the aerodynamic derivatives of a section model� M Eng

Thesis� University of Western Ontario� Canada� �� p

Part III�A Aerodynamic admittance of a deck segment ��

Part III

Applications

Part III�A

Experimental determination of the

aerodynamic admittance of a bridge deck

segment




Published in Journal of Fluids and Structures ��

Abstract

The gust loading on bridge decks is described by the dynamic forces on a chordwisestrip and by the spatial distribution of these forces across the span� An experimentalmethod to evaluate the aerodynamic admittance of a segment of a bridge deck thatincludes a combination of the crosssectional admittance and the spatial distributionof the forces is presented in this paper� The method is based on wind tunnel testsin turbulent ow on a motionless section model of the deck� The approach has beenvalidated experimentally on a closedbox girder bridge deck but can be applied tobridge decks of any crosssection�

� Introduction

The spectrum of the modal lift forces due to the bueting action of the wind on abridge deck has been expressed �Davenport �� by

SFz�f�

j � �

�� V B

�

�� C�

zSu�f�� ' C �

z�Sw�f

��jAz�f

��j� j Jz�f�j � j��


where f�j is a reduced frequency � fjB� �V associated with the jth vibration mode$jAz�f

��j� is a lift crosssectional admittance linked to the longitudianl� u� and vertical� w� components of the turbulence$ j Jz�f�j � j is the joint acceptance functionof mode j$ �� B� �V � Cz and C �

z are respectively the air density� the deck width� themean wind velocity at deck level� the lift coe�cient and the variations of the liftcoe�cient with angle of wind incidence$ and Su�w�Fz denotes power spectral densityof the wind components u or w or of the lift force Fz� Equation �� can be writtenfor the lateral and torsional degrees of freedom by replacing subscript z by x andm respectively�

The crosssectional aerodynamic admittance can either be approximated as inLiepmann �� Davenport �� Irwin �� using for example analytical expressions derived for a thin airfoil �Fung �� or measured as in Lamson ��Holmes �� Walshe - Wyatt �� Jancauskas �� Jancauskas - Melbourne �� Kawatani - Kim �� Sankaran - Jancauskas �� Larose�� Sato et al� �� and Bogunovic Jacobsen �� or evaluated indirectly�Grillaud et al� �� Liepmann�s approximation to Sears� function is the mostcommonly used form of the lift aerodynamic admittance of a thin airfoil in fullycorrelated gusts with sinusoidal uctuations �Liepmann ��

j �z�f�� j��

� ' ��f��

The joint acceptance function is of the form�

jJz�f�j �j� �

Z l

�

Z l

�

SL�L��"y� f��

SL�f��j�y�� j�y�� dy�dy��

where �j is the jth mode shape and SL�L��SL is the normalized crossspectrum ofthe lift force between strips � and � separated by a spanwise distance "y�

It has been shown �Larose ��$ Bogunovic Jacobsen �� and in Part I and IIof this thesis� that the evaluation of the joint acceptance function is problematicgiven the di�culty in de�ning the spatial distribution of the aerodynamic forcesthat appeared to be better correlated than the wind uctuations of the incident ow� In Part IIB an analytical model of the spanwise lift force coherence has beenproposed and compared to direct measurements of the gust loading �Larose et al�� for a family of streamlined bridge decks� The applicability of this analyticalmodel and of an ad hoc empirical model is limited until now to deck crosssectionsthat have a relatively long� fully reattached ow region� i�e� bridge decks withaerodynamic characteristics approaching the characteristics of a thin airfoil�

This paper presents an experimental method to evaluate� for bridge decks of anycrosssection� an aerodynamic admittance that includes a combination of the crosssectional admittance and the spanwise distribution of the forces� This quantity willbe referred to as segmental admittance of a motionless bridge deck� Even though it


is obtained from an intrinsically twodimensional approach �a �D section model��it has a threedimensional ��D� character when compared to the crosssectionaladmittance of a strip that is purely two dimensional ��D�� The main advantageof the proposed approach is that it eliminates the di�cult task of measuring thespanwise cohrence of the aerodynamic forces to obtain a clear picture of the spatialdistribution of the gust loading�

By itself the proposed technique to measure the aerodynamic admittance is notnew� It has been used by several researchers� e�g� Walshe - Wyatt �� Sato et

al� �� and Bogunovic Jacobsen �� However� the de�nition of what onereally measures is original and has been veri�ed experimentally� The veri�cationwas made possible with the help of the analytical and empirical models presentedin Part IIB of this thesis and published in Larose - Mann ��

� Description of the approach

The experimental technique consists of measuring the vertical� torsional and lateralforces at the extremities of a section model of a bridge deck� in turbulent ow� themodel being restrained from any windinduced motion� For the experiments to bevalid� four main criteria have to be respected�

�i� the section model has to be as rigid and as light as possible so that its lowesteigen frequency corresponds to a region of the wind spectrum where only afraction of the wind energy of the largest gusts is present �above �� Hz�$

�ii� the lengthtowidth ratio of the model should be larger than �� to ensure anadequate representation of the gust loading$

�iii� the geometric scale of the model should be selected in relation to the lengthscale of the incident turbulent ow �eld� and the spanwise coherence of the ow �eld should be representative of fullscale conditions$ for lift and pitching moment it is essential to respect in model scale the ratio of the lengthscales of the vertical component of the turbulence to the deck width� while ade�nite mismatch between the fullscale and modelscale length scales of thelongitudinal component would be acceptable$ for drag� a mismatch of � to � inucomponent length scales is acceptable since the characteristic dimensions�the deck depth �typically �� m�� is often much smaller than the ucomponentlength scales �typically �� m��$

�iv� and� the model should not undergo any visible motions during the wind tunneltests �high frequency vibrations with amplitude less than �� mm��

These four criteria can generally be met in today�s wind tunnel operations� However� since it is di�cult to alter the spanwise coherence of the incident ow �eld in


the wind tunnel to match the ow conditions of the natural wind at a given site� thataspect of criterion �iii� can be limited to a documentation of the spanwise coherence of the ow� Generally� the root coherence in the wind tunnel should approachthe root coherence of the natural wind� and if anything� it would be slightly largerin model scale� The in uence of the variations of the root coherence of the ow onthe spatial distribution of the wind loading in model scale is a �eld of research thatawaits development�

Also related to criterion �iii�� a series of experiments conducted by the author andreported in Larose �� and in Part IIA have shown that the lift and pitchingmoment crosssectional aerodynamic admittances were directly proportional to theratio of the wcomponent length scales to the deck width� Lw�B� These experiments were conducted for closedbox girder bridge decks� It was observed howeverthat the lift and pitching moment admittances were insensitive to the ucomponentlength scales� It is believed that the generation of lift and pitching moment on abridge deck is only slightly in uenced by the energy distribution of the ucomponentspectrum� This in uence is mostly associated with the energy content of the smallscale turbulence which wind tunnels have generally no problem producing�

The measured time histories of the aerodynamic forces are converted to powerspectral densities of the bodyforce coe�cients� SCz�x�m�f�� The spectra of the forcecoe�cients will in most cases show a resonant peak at the eigen frequency of theforce balance and model ensemble� If the model meets criterion �i�� the resonantpeak can be �ltered out by �tting a single degreeoffreedom mechanical admittancefunction to the peak and subsequently removing its contribution without aectingthe lower frequency part of the spectra that contains the information required�

The resulting lift force spectrum corresponds in all points to the spectrum de�nedby equation �� but in a dimensionless form

SCz�f�

j � �SFz�f

�j ��

� V �B�

��

Rearranging �� the lift aerodynamic admittance can be obtained by a quotientof a combination of spectral functions�

j Az�f�� j��

�V �SCz�f��

�C�zSu�f

�� ' C �z�Sw�f��

�j Jz�f�� j�

� ��

The subscript j has vanished from equation �� since no motion of the model shouldbe present �criterion �iv��

Similarily� the aerodynamic admittance of the pitching moment can be expressedby

j Am�f�� j��V �SCm�f

��C�

mSu�f�� ' C �

m�Sw�f��

�j Jm�f�� j�

� ��


where

SCm�f�

j � �SM�f�j �� V �B�

�

��

and M is the pitching moment about the bridge longitudinal axis�

All quantities of the righthandside of equations �� and �� can be determinedexperimentally� with the exception of the joint acceptance function� The onepointspectrum and the spanwise coherence of the wind have to be determined without the model in place and the static coe�cients and their linearized slope haveto be obtained from tests in turbulent ow for wind conditions �wind speed andturbulence intensity� identical to the test conditions that prevailed during the forcemeasurements�

Equations �� and �� can de�ne two quantities depending on how the jointacceptance function is evaluated� If j Jz�m�f�� j� is evaluated using the spanwise cocoherence of the aerodynamic forces obtained from experiments or from theempirical model of Part IIB� equations �� and �� would de�ne a crosssectionaladmittance� j Az�m�f�� j��D� roughly comparable to Sears� function ��

If j Jz�m�f�� j� is evaluated on the basis of the strip assumption using the spanwise cocoherence of the incindent w uctuations� equations �� and �� would de�nea segmental admittance� j Az�m�f�� jseg�� that includes in its de�nition the threedimensionality of the wind loading� implying a larger spanwise cocoherence of theaerodynamic forces�

As mentioned above� the evaluation of j Az�f�� jseg� has a major advantage over

the evaluation for the crosssectional admittance since it does not require an evaluation of the spanwise coherence of the forces� It can thus be used for any crosssectional shape that could be modelled by a �D section model� Its disadvantageis that it can not really be compared to any other benchmark quantity unless anevaluation of the spatial distribution of the forces is made �or is available� for thecrosssection studied�

� Experimental verication

The technique described above was used to determine the crosssection admittanceand the segmental admittance of a closedbox girder bridge deck� The results werecompared with the crosssectional admittance measured directly on a chordwisestrip of a section model� as described in Part IIA and Larose et al� �� for asimilar ratio of the turbulence length scale to the deck width and similar turbulenceintensity�

�Note that Sears� function represents the lift admittance of a thin airfoil in a fully correlatedsinusoidal gusts while j Azf

�� j��D is the lift admittance of a blu� body in turbulent �ow withrandom �uctuations in u� v and w�


�� Force measurements

A section model of the H�oga Kusten Bridge in its construction stage con�guration��( porous railings� no median divider� was mounted rigidly in the force balancerig of DMI�s �Danish Maritime Institute� �� m wide� �� m high and �� m longBoundary Layer Wind Tunnel �� The model length was l�� m and was built at ageometric scale of �� deck width� B�� m�� A sketch of the deck crosssectionis given in Figure �� The deck width to deck depth ratio for this bridge is B�D��

Figure �� Sketch of the deck crosssection of the H�oga Kusten Bridge �dimensionsin meters��

The wind tunnel tests were conducted at a mean wind speed of �� m)s in turbulent ow� Large spires mounted at the inlet of the wind tunnel� �� m upstreamof the model were used to generate the turbulent ow �eld� The vertical turbulenceintensity at deck level� Iw was �� ( and the vertical turbulence macroscale� Lw

�inverse of the wave number corresponding to the peak of the wind spectrum in itsfSw�f� representation

�� was �� m� The ratio Lw�B was �� In itself the ratioof �� does not meet criterion �iii� for this bridge deck� A ratio of Lw�B between� and �� would be a better modelling of the fullscale conditions� However� thepoint of this experimental veri�cation is to compare the present method with amore involved method where direct measurements of the spatial distribution of thewind loading have been made for many Lw�B ratios� including �� and ��for bridge decks of similar crosssection to the deck studied here �see Part II�� Inthe latter� the crosssectional admittance was found to be proportional to Lw�B tothe �)� power�

The autospectra of the u and w components of the wind are given in Figure ��

�For the von K�arm�an spectrum� Lu�w is related to the integral length scales Lxu�w through thefollowing� Lxu � ��Lu and Lxw � ��Lw �


The mean force coe�cients and their variations with angle of wind incidence weremeasured for the same exposure and are reported in Table ��

Table �� Static coe�cients for the H�oga Kusten Bridge �construction stage� inturbulent ow� The coe�cients are normalized by the deck width B�

Cz�� C �

z�� Cm�� C �

m��

��

Time histories of the drag and lift forces and the pitching moment were measuredat the extremities of the model and were recorded at a sampling frequency of ��Hz� for �� sec� Figures � and � show the power spectral densities ��pointfast Fourier transforms� of the lift and torsional aerodynamic coe�cients� Alsoshown on the graphs is a �t of a single degreeoffreedom mechanical admittancefunction of the measured spectral estimates and the resulting �ltered spectra wherethe resonant peak has been removed�

0.01 0.1 1

0.01

0.1

fB/V

f S_u

,w(f

)

u

0.01 0.1 1

0.01

0.1

fB/V

w

Figure �� Spectra of the u and w components of the incident turbulent ow as afunction of reduced frequency�


10−2

10−1

100

10−7

10−6

10−5

10−4

10−3

10−2

10−1

fB/V

S_C

z(f)

Figure �� Spectra of the lift coe�cient as a function of reduced frequency� Dots�measured spectral estimates$ light solid line� �t of a mechanical admittance function$ solid line� �ltered spectra�

10−2

10−1

100

10−8

10−7

10−6

10−5

10−4

fB/V

S_C

m(f

)

Figure �� Spectra of the pitching moment coe�cient as a function of reduced frequency� Dots� measured spectral estimates$ light solid line� �t of a mechanicaladmittance function$ solid line� �ltered spectra�


�� Evaluation of the joint acceptance function

Since the section model was motionless during the force measurements� i�e� ��y� �� in �� the joint acceptance function can be simpli�ed to a double integrationacross the model span of the cocoherence of the aerodynamic forces or of the wind uctuations�

The spanwise normalized cospectrum of the wind uctuations for the turbulent ow �eld of this investigation was described in Part IIA by

cocohw�� exp %�c��c� & cos�c��

where

� � k�"y

s� '

�

�k�L��

with k� � ��f� �V �wave number�� Lcoh�� m �a length scale �tted to the experiments�� c�� c�� and c�� is the von K�arm�an collapsing parameter�

An empirical formulation of spanwise cocoherence of the lift and torsional forcesfor a family of closedbox girders similar to the deck of the H�oga Kusten Bridge wasgiven in Part IIB and is of the form�

cocohL�M �� exp %�c��c� & cos�c��

where� for lift�

� � ka�"y

s� '

�

�ka�LF ��$ ��

LF � L�p' y

B ��

�q ' r yB ��

p � �� q � �� r � �� L � ��m$ ��

a �

�B

D

�� p' yB ��

�q ' r yB ��

p � �� q � �� r � ��$ ��

and c�� c�� and c��

For the pitching moment�

a �

�B

D

�� p' yB ��

�q ' r yB ��

p � �� q � �� r � ��

and c�� c�� and c��

The joint acceptance function was calculated for the three cases given above andthe results are shown in Figure �� The dierence between the curves for the forcesand the curve for the wind is due to the threedimensionality of the wind loading�the spanwise cocoherence of the forces being larger than the cocoherence of theincident ow�


0.001 0.01 0.1 10.01

0.1

1

fB/V

Join

t acc

epta

nce

func

tion

lift and torsional forces

w component

B=0.367 m, span=2.55 m

Figure �� Variations of the joint acceptance function with reduced velocity for a�� m long section model� with uniform mode shape� B�� m and B�D��

�� The aerodynamic admittances

The crosssectional admittance and the segmental admittance were calculated usingequations �� and �� for both lift and pitching moment and the results are shownin Figures � and �� The crosssectional admittance obtained with this techniqueagreed very well with the crosssectional admittance measured directly on a chordwise strip �dotted line on the graphs� for Lw�B� �� on a similar crosssection�The segmental admittance showed larger values than the crosssectional admittancesince it included the contribution of the secondary cross ow over the �� m spanthat increased the force cocoherence�

Also shown in Figures � and � is a comparison of the crosssectional admittanceobtained from the technique described here� compared to the Liepmann�s approximation to Sears� function and to the empirical model of the �D aerodynamic admittance given in Part IIB� The Sears� function considerably overestimated the �Dadmittance for fB�V smaller than �� while the empirical model gave satisfactoryresults for both lift and pitching moment admittances for this Lw�B ratio�


0.01 0.1 10.01

0.1

1

Aer

odyn

amic

Adm

ittan

ce

a)

0.01 0.1 10.01

0.1

1

fB/V

Aer

odyn

amic

Adm

ittan

ce

b)

Figure �� Variations of the lift aerodynamic admittance as a function of fB�V �On a�� dots� admittance directly measured from a chordwise strip as reported inLarose et al� �� for Lw�B��$ solid line� segmental admittance$ dashedline� crosssectional admittance determined with the present method� On b�� lightsolid line� Sears� function$ dark solid line� empirical model of �D aerodynamicadmittance of Larose - Mann �� for Lw�B��$ dashed line� crosssectionaladmittance determined with the present method�


0.01 0.1 10.01

0.1

1A

erod

ynam

ic A

dmitt

ance

a)

0.01 0.1 10.01

0.1

1

fB/V

Aer

odyn

amic

Adm

ittan

ce

b)

Figure �� Variations of the pitching moment aerodynamic admittance as a function of fB�V � On a�� dots� admittance directly measured from a chordwise stripas reported in Larose et al� �� for Lw�B��$ solid line� segmental admittance$ dashed line� crosssectional admittance determined with the present method�On b�� light solid line� Sears� function$ dark solid line� empirical model of �Daerodynamic admittance of Larose - Mann �� for Lw�B��$ dashed line�crosssectional admittance determined with the present method�


� Truss girder decks and other blu� cross�sections

The present method can as well be applied to bridge decks with more complexcrosssections such as truss girder decks or composite decks made of a concreteslab supported by longitudinal edge beams and transversal oor beams� The maindi�culty of the technique resides in building a section model with a very large exural and torsional rigidity to reduce the in uence of the resonant ampli�cation ofthe aerodynamic forces� This technique has recently been applied for a truss girderbridge deck with fairly large structural members and the results are presented inFigure �� The static aerodynamic force coe�cients of the deck in question are givenin Table � �based on the deck width��

The drag admittance was calculated by replacing subscript z by subscript x inequation �� The results are compared in Figure � to the empirical expression ofthe drag admittance of at plates normal to a turbulent ow as proposed by Vickery��

Table �� Static aerodynamic force coe�cient for a truss girder bridge deck inturbulent ow �normalized by the deck width�

Cz�� C �

z�� Cm�� C �

m�� Cx�� C �

x��

��

The measured admittance of Figure � agreed well with similar measurementsmade for the truss girder of the AkashiKaikyo Bridge reported by Sato et al�

��


0.01 0.1 10.01

0.1

12

5A

erod

. adm

it.

Sears

fB/V

a) lift

0.01 0.1 1

1

2

5

10

20

50

Aer

od. a

dmit.

x C

z´^2

fB/V

Sears

b) lift

0.01 0.1 10.01

0.1

125

102050

fB/V

Aer

od. a

dmit.

Sears

c) moment

0.01 0.1 10.1

1

fD/V

Aer

od. a

dmit.

d) dragVickery

Figure �� Segmental aerodynamic admittance for a truss girder bridge deck measured with the present technique� On b� the ordinate has been multiplied by thelift slope squared� On d� the dashed line refers to the drag admittance proposedby Vickery �� %�� ' ��fD�V ��&� where D is the deck depth�

Conclusion

An experimental approach to evaluate the aerodynamic admittance of a segmentof a bridge deck was presented� The aerodynamic characteristic obtained with thistechnique has a threedimensional character since it includes the in uence of thespanwise distribution of the aerodynamic forces� The approach can be used forbridge deck of any crosssection�


References

Bogunovic Jakobsen� J� �� Fluctuating wind load and response of a linelikeengineering structure with emphasis on motioninduced wind forces� Ph�D�Thesis� The Norwegian Institute of Technology� Trondheim� Norway�

Davenport� A�G� �� The response of slender� linelike structures to a gustywind� In Proceedings of Institution of Civil Engineers ��

Fung� Y�C� �� An Introduction to the Theory of Aeroelasticity� New York�Dover Publications�

Grillaud G�� Flammand O� � Barr�e C� �� Comportement au vent du Pontde Normandie� Etude en sou.erie atmosph�erique sur maquette a�ero�elastique�a �echelle du �)��eme� Centre Scienti�que et Technique du B�atiment� Nantes�France� ENAS �� C� �� p�

Holmes J�D� �� Prediction of the response of a cablestayed bridge to turbulence� In Proceedings of �th International Conference on Buildings andStructures London� England� Cambridge� Cambridge University Press�

Irwin H�P�A�H� �� Wind Tunnel and Analytical Investigations of the Responseof Lions� Gate Bridge To Turbulent Wind� National Research Council ofCanada� NAE LTRLA�� p�

Jancauskas E�D� �� The crosswind excitation of blu structures and the incident turbulence mechanism� Ph�D� Thesis� Monash University� Melbourne�Australia�

Jancauskas E�D� � Melbourne W�H� �� The aerodynamic admittance oftwodimensional rectangular cylinders in smooth ow� Journal of Wind En�gineering and Industrial Aerodynamics ��

Kawatani M� � Kim H� �� Evaluation of aerodynamic admittance for bueting analysis� Journal of Wind Engineering and Industrial Aerodynamics ��

Lamson P� �� Measurements of lift uctuations due to turbulence� NACA TN��

Larose� G�L� �� The response of a suspension bridge deck to turbulent wind�the taut strip model approach� M�E�Sc� Thesis� University of Western Ontario� London� Ontario� Canada�

Larose� G�L�� Tanaka H�� Gimsing N�J� � Dyrbye C� �� Direct measurements of bueting wind forces on streamlined bridge decks� Journal of WindEngineering and Industrial Aerodynamics ��

Larose G�L� � Mann J� �� Gust loading on streamlined bridge decks� Jour�nal of Fluids and Structures ��

Liepmann� H�W� �� On the application of statistical concepts to the buetingproblem� Journal of Aeronautical Science ��


Sankaran� R� � Jancauskas� E�D� �� Direct measurement of the aerodynamic admittance of twodimensional rectangular cylinders in smooth andturbulent ows� Journal of Wind Engineering and Industrial Aerodynamics��

Sato H�� Matsuno Y� � Kitagawa M� �� Evaluation of the aerodynamic admittance for the stiening girder of the Akashi Kaikyo Bridge� In Proceedingsof the � th� Japan National Symposium on Wind Engineering� paper �� in japanese��

Vickery� B�J� �� On the ow behind a coarse grid and its use as a model ofatmospheric turbulence in studies related to wind loads on buildings� NationalPhysical Laboratory� Aero Report ��

Walshe D�E� � Wyatt T�A� �� Measurements and application of the aerodynamic admittance function for a boxgirder bridge� Journal of Wind Engi�neering and Industrial Aerodynamics ��

Part III�B Performance of streamlined bridge decks ��

Part III

Applications

Part III�B

Performance of streamlined bridge decks in

relation to the aerodynamics of a �at plate


and Flora M� Liveseya



Published in J� of Wind Engineering and Industrial Aerodynamics ��

Abstract

The aerodynamics of three modern bridge decks are compared to the aerodynamicsof a �� at plate� The comparisons are made on the basis of the analyticalevaluation of the performance of each crosssection to the bueting action of thewind� In general the closedbox girders studied in this paper showed buetingresponses similar to a at plate with the exception of the multibox girder whichperformed much better aerodynamically�

� Introduction

The objective of this paper is to compare the aerodynamic performance of selectedstreamlined bridge decks to the performance of a at plate in turbulent ow� Thecomparisons are based on the main aerodynamic properties of the deck crosssection�that is� static force coe�cients� aerodynamic derivatives and aerodynamic admittances� Most of the properties were obtained from wind tunnel tests on sectionmodels carried out at the Danish Maritime Institute�

The deck cross sections studied here� all closed steel box girders that have beendesigned with the emphasis on aerodynamics �therefore the appellation streamlined��

�� G�L� Larose and F�M� Livesey

belong to the following structures� the proposed suspension bridge across the Straitsof Messina� Italy$ the Pont de Normandie� recently completed in France$ and theH�oga Kusten Bridge under construction in Sweden�

Fig� � shows the crosssections of the three abovementioned bridge decks� Theproposed Messina Straits Bridge would have a ��m long mainspan� a deck widthof ��m and a widthtodepth ratio� B�D� of �� The Pont de Normandie� an ��mmainspan cablestayed bridge� has a B�D � �� a ��m wide deck� once the nosingswere in place� with a depth of �m�� The H�oga Kusten Bridge� a ��m mainspansuspension bridge� has a deck width of ��m and a maximum depth of �m yieldingB�D � �� The reference at plate is very slender with a B�D � �� Note thatboth the Pont de Normandie and the H�oga Kusten Bridge were tested in theirconstruction stage con�guration with temporary railings�

Figure �� Deck crosssections of the Messina Straits Bridge� Pont de Normandieand the H�oga Kusten Bridge �dimensions in meters��

The evaluation of the performance of each of the decks in comparison to a atplate is based on the analytical prediction of the bueting response� Similar studiescan be found in the literature %�� & but in most of the cases the comparisonsare based on the aeroelastic behaviour and the stability limit� except in %�& wherethe experimentally determined responses of various bridge decks �from the TacomaNarrows to the Pont de Normandie� are compared to the response of a at plate�

� Comparison of static force coe�cients

Section model tests were conducted for all three bridge decks in DMI�s ��mwideboundary layer wind tunnel or in the ��mwide closed circuit wind tunnel� to mea


sure time averaged drag� lift and moment coe�cients versus angle of wind incidence�The tests were typically conducted in turbulent ow with a vertical turbulence intensity of �( generated by large spires ��m upstream of the model� with the exceptionof the Messina Bridge which was tested in smooth ow� The coe�cients for the atplate in turbulent ow were obtained from %�&�

The coe�cients presented in Fig� � were normalized by B and were convertedto force coe�cients Cx� Cz and Cm� with reference to a coordinate system �xed tothe body� where the subscripts x� z�m denote respectively the lateral� vertical andtorsional degrees of freedom of the deck� The table in Fig� � summarises the resultsfor �� wind incidence$ C �

z and C �m denote the rate of change of the coe�cients with

angle of attack around ��

Figure �� Variations of Cx� Cz and Cm with angle of wind incidence for � crosssections�

Assuming that the instantaneous forces on the bridge deck are equal to the steadyforces induced by a steady wind of the same relative velocity and direction as theinstantaneous wind �i�e� quasisteady aerodynamics�� Davenport %�& showed that�

�� the forces due to turbulence are proportional to the product of the mean


wind velocity� the longitudinal velocity uctuations and Cx�z�m� added to theproduct of the vertical velocity uctuations and C �

x�z�m$

�� Cx and C �z� contribute respectively to the lateral and vertical aerodynamic

damping by an amount proportional to the mean wind velocity$

�� C �m contributes to the torsional stiness of the system$ if positive� the torsional

stiness will decrease in proportion to the square of the wind velocity�

Thus� Fig� � reveals a good deal about the aerodynamic performance of thedecks studied� First both C �

z and C �m are positive for all crosssections which means

favorable aerodynamic damping� In general the decks have C �m values that follow the

at plate value and a C �z that is relatively smaller than that for the at plate� Also�

they have a negative Cz at �� which indicates that the deck is pushed downwardas the wind speed increases� This negative lift is generally compensated for by theupward force produced by the wind on the cable system�

The multibox girder of the Messina Straits Bridge� however� has very favorablestatic coe�cients that put it in a class of its own� This deck has both C �

z and C �m

equal to a quarter of the corresponding values of the other crosssections and Cz�m

are almost zero at �� Good performance with regards to bueting is anticipatedas well as a high stability limit�

� Aerodynamic derivatives and admittances

The motional aerodynamic derivatives and the aerodynamic admittances have beenmeasured directly or indirectly for these deck crosssections� The main characteristics are presented here with the emphasis put on the relationships between thesequantities and the static coe�cients�

The aerodynamic admittance is a measure of the eectiveness of a body in extracting energy from the oncoming turbulence� It was introduced by Davenport��rstly to express that the wind loading is not necessarily quasisteady and may varywith frequency� and secondly to represent the spatial variations in the ow over theregion in uencing the forces on a crosssection� This second aspect is now beingstudied in more details by the authors %�&�

Fig� � presents direct measurements �dashed lines� of aerodynamic admittancesas a function of reduced frequency �f� � fB�V �� The solid line represents theLiepmann�s approximation to the Sears function analytically derived for a at plate�thin airfoil� with a lift slope of �� in a fully correlated sinusoidal gust %�&�

j��f��j� ��

�� ' ��f��Liepmann� ��

The admittance measurements were done either by integration of the surface pressures on a crosssectional strip of a section model �as in Part IIA� or by measuring


10−2 10−1 100

10−1

100

Aer

o. a

dmitt

ance

Hoga Kusten, lift

10−2 10−1 100

10−1

100

Hoga Kusten, torsion

10−2 10−1 100

10−1

100

fB/V

Aer

o. a

dmitt

ance

Normandie, lift

10−2 10−1 100

10−1

100

fB/V

o: Messina; X: Flat Plate

lift

Figure �� Variations of aerodynamic admittances with reduced frequency� Solid

line� Liepmann�s approximation� equation ��$ dashed lines� direct measurementsof admittance on section models$ and light lines with o and x� indicator ofunsteadiness i�e� ratio of the unsteady lift slopes C �

z�f�� to the steady case C �

z��

the lift forces at the extremity of a very rigid ��m long section model and dividingby the incoming ow characteristics �autospectra and spanwise coherence�� Theprocedures behind the measurements are reported elsewhere %�� & �see also PartIIA and IIIA��

Also shown on Fig� � �o and x� is an attempt to relate the aerodynamic admittanceto the ratio C �

z�f��C �

z�f � �� that is� the unsteady lift slope to the steady case�Intuitively this ratio should be related to the �rst de�nition of the aerodynamicadmittance� The unsteady lift slope can be directly obtained from the relationshipsbetween the force coe�cients and the aerodynamic derivatives H�

� and A�� %�& as per

equations �� and �� obtained assuming quasisteady aerodynamics�

H�

� ��C �

z

�K$ A�

� �C �m

�K$ H�

� ��C �

z

�K�$ A�

� �C �m

�K��

H�

� �C �z

�Kn� $ A�

� ��C �

m

�Kn� $ nz�K� �

A��

H��

$ n��K� �A��

H��

��


where K � ��fBV � B is the deck width� V the mean wind speed and nz�� are the

points of application of the aerodynamic forces expressed as a fraction of B�For the at plate� the unsteady lift slope can be obtained analytically via the

relationship between the Theodorsen function %�&� the aerodynamic derivatives andequations �� The Sears function can also be related to the aerodynamic derivativesthrough the Theodorsen function %�&�

The ratio C �z�f

��C �z�� can be seen as an indicator of unsteadiness� If it drops

with increasing f�� it is believed that the aerodynamic admittance will do the same�This trend can be observed� at least qualitatively� on Fig� � for the H�oga KustenBridge and the Pont de Normandie�

Based on this� it is expected that the aerodynamic admittance for the MessinaBridge would not drop with reduced frequency but increase to values above unity�This would correspond to the eect of the selfinduced turbulence created by theupwind box girder and aecting the wind loading on the downstream componentsof the deck at higher reduced frequencies�

0 5 10 15 20−8

−6

−4

−2

0

H1s

tar

Hoga Kusten Bridge

0 5 10 15 200

1

2

3

A1s

tar

Hoga Kusten Bridge

0 2 4 6 8 10−1

−0.5

0

0.5

H2s

tar

0 2 4 6 8 10−0.3

−0.2

−0.1

0

A2s

tar

0 2 4 6 8 10−6

−4

−2

0

Reduced Velocity

H3s

tar

0 2 4 6 8 100

0.5

1

1.5

Reduced Velocity

A3s

tar

Figure �� Variations of H�� and A�

�� with reduced velocity for the H�oga KustenBridge� The dashed lines are predictions derived from quasisteady aerodynamicsusing the coe�cients of Fig� � and n � ��


Fig� � to � shows some of the measured motional aerodynamic derivatives in Scanlan�s notation %�&� using the inital displacement method� as a function of reducedvelocity� and compares them to approximations using equations �� and �� Theagreement is surprisingly clear except forH�

� andA�� where quasisteady aerodynam

ics would predict erroneous values� This could lead to miscalculation of the torsionalaerodynamic damping if quasisteady aerodynamic assumptions were maintained�The relationship between H�

� � A�� and the lift and torsion slopes is questionable and

the reader is referred to %��& for a detailed discussion on the matter�

� Prediction of the bu�eting response

The spectral approach described in %��& was used here to predict the bueting response of the cross sections� In the calculations� equation �� was used along withthe following functions� aerodynamic admittance� jAz�f

��j� as measured �exceptfor Messina and the at plate where the Liepmann�s approximation to the Searsfunction� equation �� was used�� aerodynamic damping extracted from the aerodynamic derivatives �� Kaimal wind spectra fSw�f��

�w� a typical �rst symmetric

mode shape� a structural damping ��s� of ��( of critical and a ratio of the mass of

0 5 10 15 20 25−2

−1.5

−1

−0.5

0

H1s

tar

Messina Straits Bridge

0 5 10 15 20 25−0.2

0

0.2

0.4

0.6

A1s

tar

Messina Straits Bridge

0 5 10 15 20 25−3

−2

−1

0

H2s

tar

0 5 10 15 20 25−0.6

−0.4

−0.2

0

A2s

tar

0 5 10 15 20 25−8

−6

−4

−2

0

Reduced Velocity

H3s

tar

0 5 10 15 20 250

0.5

1

1.5

2

Reduced Velocity

A3s

tar

Figure �� Variations ofH�� andA

�� with reduced velocity for the Messina Straits

Bridge� The dashed lines are predictions derived from quasisteady aerodynamicsusing the coe�cients of Fig� � and n � ��


0 5 10 15 20−8

−6

−4

−2

0

H1s

tar

Pont de Normandie

0 5 10 15 200

0.5

1

1.5

2

A1s

tar

Pont de Normandie

0 2 4 6 8 10−0.4

−0.2

0

0.2

Reduced Velocity

H2s

tar

0 2 4 6 8 10−0.6

−0.4

−0.2

0

Reduced Velocity

A2s

tar

Figure �� Variations of H�� and A�

�� with reduced velocity for the Pont de Normandie� The dashed lines are predictions derived from quasisteady aerodynamicsusing the coe�cients of Fig� � and n � ��

the displaced air to the mass of the deck� �B��m� corresponding to the characteristics of each of the deck� This ratio was in fact the same for Pont de Normandie�H�oga Kusten and the at plate�

The normalized modal rootmeansquare response �acceleration�� for the verticalmodes of the deck can be expressed by�

rj�zIwB�

j

� C �z

��

V

fjB

��B�

m

sfSw�f�

�wjAz�f��j� jJz�j�f��j� ��

�s ' �a�z�f��

where the subscript j denotes the jth mode of vibration and jJz�j�f��j� is the jointacceptance function which was calculated on the basis of the strip assumption� Asimilar expression can be written for the torsional response by replacing subscriptz by m and �B��m by �B��I�� I� being the mass moment of inertia per unit lengthof the deck�

The aerodynamic damping was de�ned as�

�a�z�f�� H�

� �f��B�

�m$ �a��f

�� A�

��f��B�

�I��

Fig� � presents the results of the predictions along with the aerodynamic derivatives used to de�ne the aerodynamic damping� The rms modal resonant response�acceleration�� rj� was normalized by the vertical turbulence intensity Iw� the angular frequency of mode j� �

j � and B as in %�� &� The vertical responses collapse wellon one line for the Pont de Normandie and H�oga Kusten and were found to perform


slightly better than the at plate at high reduced velocities� This dierence canmostly be attributed to the dierences in the low reduced frequency range betweenthe Sears function and the directly measured aerodynamic admittances� The deckof Messina showed a much better performance�

0 5 10 15 20−8

−6

−4

−2

0

H1s

tar

a) aerodynamic derivatives, H1*

0 2 4 6 8 10−0.5

−0.4

−0.3

−0.2

−0.1

0b) aerodynamic derivatives, A2*

A2s

tar

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35c) vertical resonant response

V/fB

r/(I

Bw

^2)

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25d) torsional resonant response

V/fB

r/(I

w^2

)

Flat Plate Normandie Hoga KustenMessina

Figure �� Variations of the aerodynamic derivatives a� H�� b� A

�� and the nor

malized resonant modal responses� rj�IwB�j � c� vertical� d� torsional with reduced

velocity V�fjB for three deck crosssections compared to a �� at plate

For the torsional response� the aerodynamic damping �which is a function ofA�� and B� appears to dominate� since the performance seemed to be inversely

proportional to the widthtodepth ratio� For example� the H�oga Kusten Bridgehas considerably lower A�

� values than the Pont de Normandie and the at plate�directly resulting in a larger torsional response� Also� it should be noted that thelow torsional response of the Messina Bridge is largely due to strong aerodynamicdamping provided by optimized wind screens that act as dampers and its very largedeck width�


Conclusions

Examining the aerodynamic properties and the bueting response of the MessinaStraits Bridge leads to the conclusion that bridge designs can be optimized to perform aerodynamically much better than a at plate� both in lift and torsional responses� However� the Pont de Normandie and the H�oga Kusten Bridge� whichare more typical deck designs� gave similar responses to a at plate according tothe bueting response predictions� The lower torsional aerodynamic damping forthe H�oga Kusten Bridge� partly due to its lower widthtodepth ratio� leads to thelarger torsional response predicted by the bueting theory�

Acknowledgements� The valuable contribution of Dr Hiroshi Tanaka to thispaper� through discussions and supervision� is kindly acknowledged as well as the�nancial support of the �rst author by the Danish Research Academy�

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